Krsna first incarnates as the three purusa-avataras, namely the Maha-Visnu or Karanodakasayi avatara, the Garbhodakasayi avatara and the Ksirodakasayi avatara. This is confirmed in the Satvata-tantra [specified as Naradiya tantra in Bhavartha-dipika 2.6.38 by Srila Sridhara Goswami, see also SB 1.3.8]:
visnos tu trini rupani
purusakhyany atho viduh
ekam tu mahatah srastr
dvitiyam tv anda-samsthitam
tani jnatva vimucyate
"For material creation, Lord Krsna's plenary expansion assumes three Visnus. The first one, Maha Visnu, creates the total material energy, known as mahat-tattva. The second, Garbhodakasayi Visnu, enters into all the universes to create diversities in each of them. The third, Ksirodakasayi Visnu, is diffused as the all-pervading Supersoul in all the universes and is known as Paramatma. He is present even within the atoms. Anyone who knows these three Visnus can be liberated from material entanglement." (Satvata-tantra quoted in Bg. 7.4 purport)
Krsna is the original Personality of Godhead. His expansion is Balarama, who expands the original catur vyuha or quadruple expansions.
1) Vasudeva 2) Sankarsana 3) Pradyumna 4) Aniruddha
These original catur vyuha expansions reside in Mathura and Dvaraka. From them twenty four forms of Visnu expand. They are named differently according to the arrangement of the conch, disc, lotus and club in Their hands. All of these twenty four forms reside in each Vaikuntha planet with the predominating Deity of that planet. From the original catur vyuha the second catur vyuha expands. In this second catur vyuha the form of Sankarsana is also called Maha Sankarsana. It is from Maha Sankarsana that Maha Visnu becomes manifested.
Original Personality of Godhead
ORIGINAL CATUR VYUHA
Vasudeva, Sankarsana, Pradyumna, Aniruddha
SECOND CATUR VYUHA
Vasudeva, Sankarsana,¿ Pradyumna, Aniruddha
(each expand into three and³then another two expansions)
1) MAHA VISNU lies on the Causal Ocean which appears in one corner of the spiritual world. He manifests the mahat tattva (or the sum total of material energy). It is into this mahat tattva that He exhales all of the seedlike universes through the pores of His skin. These seedlike universes then expand as the different material elements form coverings around them. Each of the coverings is ten times thicker than the previous covering and form a shell-like covering. When Maha Visnu impregnates the living entities by His glance into the material nature it begins to manifest its various energies.
2) When the universes have thus developed in the womb of material nature, Maha Visnu expands as GARBHODAKASAYI VISNU, Who enters into each universe with the living entities of that particular universe and thus activates each universe by His presence. He lies down on Ananta Sesa who lies on the Garbha ocean, which is the perspiration from the Lord's body and half fills the universe. From His navel comes a lotus bud which is the total form of the living entities' fruitive activity. The lotus grows dissipating the darkness of the universe. On top of the lotus Brahma, the first living being appears. Situated on the lotus, Brahma could not understand anything. He began entering the stem and climbed down to find its origin. Not finding anything, Brahma again returned to the top of the lotus where he heard the word ta-pa (austerity). Hearing the sound Brahma underwent penance for one thousand celestial years; (6x30x12x1000 earthly years). Being very pleased with Brahma's tapasya, the Lord manifested the Vaikuntha planets to him. Seeing Vaikuntha, Brahma became very happy and bowed to the Lord. Being very pleased with Brahma the Lord shakes his hand and reveals how to create the universe.
3) To maintain the universe Garbhodakasayi Visnu expands as KSIRODAKASAYI VISNU, Who is the all-pervading Supersoul (Paramatma). By His entering into every atom He maintains the whole universe. His abode is Svetadvipa, an island in the ocean of milk.
Krsna's energies can also be divided into three: His energy of thinking feeling and acting. When He exhibits His thinking energy, He is the Supreme Lord; when He exhibits His feeling energy, He is Lord Vasudeva; when He exhibits His acting energy, He is Sankarsana Balarama. Without His thinking, feeling and acting, there would be no possibility of creation. Although there is no creation in the spiritual world - for there the planets are beginningless - there is creation in the material world. In either case, however, both the spiritual and material worlds are manifestations of the energy of acting, in which Krsna acts in the form of Sankarsana and Balarama.
References: SB 1.3.1-5, 2.9.4-9, 3.8.11-21, 10.46.31
Lila-avataras (pastime avataras)
During Brahma's one day there are twenty-five Lila-avataras also known as Kalpa avataras because they appear in every Kalpa. Out of these, the incarnation of Hamsa and Mohini are not permanent, but Kapila, Dattatreya, Rsabha, Dhanvantari and Vyasa are five eternal forms, and they are more celebrated. The incarnations of the tortoise Kurma, the fish Matsya, Nara-narayana, Varaha, Hayasirsa, Prsnigarbha, and Balarama are considered to be vaibhava-avataras.
1) Catuhsana - The four Kumaras, sons of Lord Brahma
They appeared in the beginning of creation and are specifically empowered to distribute transcendental knowledge. They are empowered with the Lord's jnana-sakti and therefore they are also known as saktyavesa-avataras.
2) Narada Muni - The son of Lord Brahma
He is empowered with the Lord's bhakti-sakti and therefore he is also saktyavesa-avatara.
3) Varahadeva - Sukara or the Boar incarnation
The Boar incarnation appeared in two different millenniums. During the period of Svayambhuva Manu, the earthly planet remained submerged in the water of devastation, so the Lord appeared as a white boar and lifted the earth and set it properly. During the period of Caksusa Manu Lord Boar was red and He killed the demon Hiranyaksa.
4) Matsya - The Fish incarnation
He appeared to show special mercy to Satyavrata Muni. After the period of Caksusa Manu when there was a partial inundation, he also protected Satyavrata Muni (who later became Vaivasvata Manu) by keeping him safe on a boat.
5) Yajna - The son of Prajapati Ruci and Akuti
During the period of Svayambhuva Manu there was no qualified Indra. So the Lord appeared as Yajna and took that position.
6) Nara Narayana - The twin sons of King Dharma and Murti
They are partial expansions of Krsna and Arjuna. Nara and Narayana Rsis exhibited the Lord's renunciation.
7) Kapiladeva - The son of Kardama Muni and Devahuti
He explained the original (personal) Sankhya philosophy. Kapiladeva exhibited the Lord's transcendental knowledge.
8) Dattatreya - The son of Atri Muni and Anasuya
He is a combined incarnation of Lord Visnu, Brahma and Siva. He spoke on the subject of transcendence to Alarka, Prahlada, Yadu, etc.
9) Hayasirsa - Lord Hayagriva
At the end of the millennium ignorance personified took the form of a demon, stole the Vedas and took them to the planet Rasatala.
At that time the Lord, at the request of Brahma, retrieved them after assuming the form of a horse.
10) Hamsa - The Swan incarnation
Due to bewilderment, Brahma, could not answer some questions put forward by his sons. At that time the Lord appeared as a swan and answered the questions.
11) Prsnigarbha - The incarnation who appeared before Dhruva
He created the planet known as Dhruvaloka for the habitation of Dhruva Maharaja.
12) Rsabha - The son of King Nabhi and Merudevi
His oldest son was Bharata Maharaja of which the earth was named after as Bharat-varsa. He instructed his sons to follow the path of perfection by tapasya (asceticism).
13) Prthu - The incarnation of the Lord's ruling force
Due to the demonic nature of King Vena, the sages killed him by a curse. They prayed for the Lord to appear and churned the two arms of the dead body according to a specific method and the Lord appeared as Prthu.
14) Nrsimhadeva - The half-man half-lion incarnation
He appeared to protect Prahlada by killing his demoniac father Hiranyakasipu. Sri Nrsimha manifests various bhavas, mostly wrathful but also peaceful ones (like some of His murtis in Ahobilam or Santa Nrsimha in Puri). Gaudiya Vaisnava siddhanta about Sri Nrsimha appearing in this universe is that He's not a form of Sri Visnu residing in one of Vaikunthas but directly Sri Krsna. His lila manifests only in the material realm, although perpetually and in that sense is also eternal (Prahlada Maharaja worships Krsna, not Narayana.
Nrsimha Purana 41.36 says that on his slate he has written the name 'Krsna' and he remembers Krsna nama.). This is hinted on in SB 3.24.33 p., last par., and BG 4.8. He personifies this BG verse. nrsimha-rama-krsnesu sad-gunyam paripuritam: "In Narasimha, Rama and Krsna, all the six opulences (strength, wealth, renunciation, splendor, energy, wisdom) are fully manifest." (Padma Purana, quoted by Srila Rupa Gosvami in Laghu-bhagavatamrta 1.5.16)
Srila Bhaktivinoda Thakura has written five beautiful prayers in "Sri Navadvipa Bhava Taranga" for receiving the mercy of Lord Nrsimha. These prayers are certainly assurance to all sincere devotees that the worship of Lord Nrsimha is purely in the line of aspiring love and devotion to Sri Sri Radha and Krsna.
e dusta hrdaye kama adi ripu chaya
kutinati pratisthasa sathya sada raya
hrdaya-sodhana ara krsnera vasana
nrsimha-carane mora ei to' kamana
Within my sinful heart the six enemies headed by lust perpetually reside, as well as duplicity, the desire for fame, plus sheer cunning. At the lotus feet of Lord Narasimha, I hope that He will mercifully purify my heart and give me the desire to serve Lord Krsna.
kandiya nrsimha-pade magibo kakhana
nirapade navadvipe jugala-bhajana
bhaya bhaya paya yan'ra darsane se hari
prasanna hoibo kabe more daya kari
Weeping, I will beg at the lotus-feet of Lord Narasimha for the benediction of worshipping Radha and Krsna in Navadvipa, perfectly safe and free from all difficulties. When will this Lord Hari, whose terrible form strikes fear into fear itself, ever become pleased and show me His mercy?
yadyapi bhisana murti dusta-jiva-prati
prahladadi krsna-bhakta-jane bhadra ati
kabe va prasanna ho'ye sa krpa-vacane
nirbhaya karibe ei mudha akincane
Even though Lord Narasimha is terrifying toward the sinful souls, He offers great auspiciousness unto the devotees of Lord Krsna headed by Prahlada Maharaja. When will He be pleased to speak words of compassion unto me, a worthless fool, and thereby make me fearless?
svacchande baiso he vatsa sri-gauranga-dhame
jugala-bhajana hau rati hau name
mama bhakta-krpa-bale vighna jabe dura
suddha cite bhajo radha-krsna-rasa-pura
He will say, "Dear child! Sit down freely and live happily here in Sri Gauranga-dhama. May you nicely worship the Divine Couple, and may you develop loving attachment for Their Holy Names. By the mercy of My devotees, all obstacles are cast far away. With a purified heart, just perform the worship of Radha and Krsna, for such worship overflows with sweet nectar."
ei boli' kabe mora mastaka-upara
sviya sri-carana harse dharibe isvara
amani jugala-preme sattvika vikare
dharaya lutibo ami sri-nrsimha-dvare
Saying this, will that Lord delightedly place His own divine lotus-feet upon my head? I will experience sublime love for the Divine Couple Radha-Krsna and undergo the ecstatic transformations called sattvika. Falling on the ground, I will roll about at the door of Sri Narasimha's temple. (Srila Bhaktivinoda Thakura, Sri Navadvipa Bhava Taranga 36-40)
Bhaktivinoda Thakura prays to Sri Nrsimhadeva to clear his path to Radha Krsna worship. He wouldn't be praying to the equivalent of an "ordinary" Visnu tattva in Vaikuntha for entrance into Goloka.
Another argument against the identity of the two Nrsimhas is that the Pancaratra texts identify the Nrsimha avatara of the dasavatara as manifesting from Sankarsana: haris sankarsanamsena narasimha-vapur dharah (Padma tantra 1.2.31 and Visnu-dharmottara Purana 3.78(2).5-7) However, the vyuha Nrsimha is manifested from Pradyumna (CC Madhya 20.206).
Related: Nrsimha murtis and lilas
15) Kurma - The Tortoise incarnation
He appeared to become the resting place of the Mandara Hill, used as a churning rod by the demons and demigods to produce nectar.
16) Dhanvantari - The father of Ayurveda
He appeared from the ocean of milk with the pot of nectar.
17) Mohini - The Lord's form of a beautiful woman
The Lord tricked the demons by appearing in the form of a beautiful woman and asked the demons for the nectar and distributed it to the demigods.
18) Vamanadeva - The dwarf incarnation, son of Kasyapa Muni and Aditi
Assuming the form of a dwarf brahmacari the Lord visited the fire sacrifice of Bali Maharaja. He begged three steps of land and with those steps He took the whole universe. He is the son of Kasyapa and Aditi.
19) Parasurama - Bhrgupati, the son of Jamadagni Muni and Renuka
He annihilated the ksatriyas twenty-one times because of their rebellion against the brahmanas.
20) Raghavendra - Lord Ramacandra, son of King Dasaratha and Kausalya
He came to perform pleasing work for the demigods, protect His devotees and kill Ravana and his followers. The Lord assumed the form of a human being and exhibited superhuman powers by controlling the Indian Ocean.
21) Vyasadeva - The son of Parasara Muni and Satyavati
He divided the one Veda into several branches and sub-branches, seeing that the people in general were becoming less intelligent.
22) Lord Balarama - The first plenary expansion of the Lord
He acts as Lord Krsna's older brother in Their pastimes in Vrndavana.
23) Lord Krsna - The original form of the Lord
Both Balarama and Krsna appeared in the family of Vrsni (Yadu dynasty) and in doing so removed the burden of the world.
24) Buddha - The son of Anjana
He appeared in the province of Gaya in the beginning of Kali-yuga to delude those who are inimical to the faithful. He also appeared to stop animal slaughter in the name of Vedic sacrifices by preaching non-violence and by not accepting Vedic authorities. There is also another Buddha incarnation mentioned in another Kali-yuga when Lord appeared to bewilder the demons who were destroying the inhabitants of other planets by flying unseen in space ships made by the demon Maya.
25) Kalki - The son of brahmana Visnu-yasa
He will appear in the village Sambhala at the end of Kali-yuga. He will mount a horse, Devadatta, and taking His sword, will kill millions upon millions of degraded barbarian humans. He will also commence Satya-yuga.
They are Brahma (rajo-guna), Visnu (sattva-guna) and Siva (tamo-guna). Brahma is one of the living entities, but due to his devotional service he is very powerful. This primal living entity, master of the mode of material passion, is directly empowered by the Garbhodakasayi Visnu to create innumerable living entities. In Brahma-samhita (5.49) Brahma is likened to valuable jewels influenced by the rays of the sun, and the sun is likened to the Supreme Lord Garbhodakasayi Visnu. If in some kalpa there is no suitable living entity capable of acting in Brahma's capacity, Garbhodakasayi Visnu Himself manifests as Brahma and acts accordingly.
Similarly, by expanding Himself as Lord Siva, the Supreme Lord is engaged when there is a need to annihilate the universe. Lord Siva, in association with maya, has many forms, which are generally numbered at eleven. (They are not, technically speaking, avataras since this term is used only for Sri Visnu's manifestations. Hanuman is an expansion of Siva, Siva-amsa. He is not a jiva.) Lord Siva is not one of the living entities; he is a transformation of Krsna Himself. The example of milk and yogurt is often given in this regard - yogurt is a preparation of milk, but still yogurt cannot be used as milk. Similarly, Lord Siva is an expansion of Krsna, but he cannot act as Krsna, nor can we derive the spiritual restoration from Lord Siva that we derive from Krsna. The essential difference is that Lord Siva has a connection with material nature, but Visnu or Lord Krsna has nothing to do with material nature. In Srimad-Bhagavatam (10.88.3) it is stated that Lord Siva is a combination of three kinds of transformed consciousness known as vaikarika, taijasa and tamasa.
The Visnu incarnation, although master of the modes of goodness within each universe, is in no way in touch with the influence of material nature. Although Visnu is equal to Krsna, Krsna is the original source. Visnu is a part, but Krsna is the whole. This is the version given by Vedic literatures. In Brahma-samhita the example is given of an original candle which lights a second candle. Although both candles are of equal power, one is accepted as the original, and the other is said to be kindled from the original. The Visnu expansion is like the second candle. He is as powerful as Krsna, but the original Visnu is Krsna. Brahma and Lord Siva are obedient servants of the Supreme Lord, and the Supreme Lord as Visnu is an expansion of Krsna.
The four yuga-avataras are also described in Srimad-Bhagavatam. In the Satya-yuga, the incarnation of God is white; in the Treta-yuga He is red; in the Dvapara-yuga, He is blackish; and in the Kali-yuga He is also blackish, but sometimes, in a special Kali-yuga, His color is yellowish as in the case of Caitanya Mahaprabhu.
This particular incarnation of the Supreme Personality of Godhead is foretold in Srimad-Bhagavatam (11.5.32):
yajanti hi sumedhasah
"In the age of Kali the Lord incarnates as a devotee, yellowish in color, and is always chanting Hare Krsna, Hare Krsna, Krsna Krsna, Hare Hare/ Hare Rama, Hare Rama, Rama Rama, Hare Hare. Although He is Krsna, His complexion is not blackish like Krsna's in Dvapara-yuga but is golden. It is in Kali-yuga that the Lord engages in preaching love of Godhead through the sankirtana movement, and those living entities who are intelligent adopt this process of self-realization."
There is no limit to the saktyavesa-avataras. The saktyavesa incarnations are of two kinds - direct and indirect. When the Lord Himself comes, He is called saksat, or a direct saktyavesa-avatara, and when He empowers some living entity to represent Him that living entity is called an indirect or avesa incarnation.
Examples of direct or saksad-avataras are the Sesa incarnation and the Ananta incarnation. In Ananta the power for sustaining all planets is invested, and in the Sesa incarnation the power for serving the Supreme Lord is invested.
Examples of indirect avataras are the four Kumaras, Narada, Prthu and Parasurama. These are actually living entities, but there is specific power given to them by the Supreme Personality of Godhead. When a specific opulence of the Supreme Lord is invested in specific entities, they are called avesa-avataras. The four Kumaras specifically represent the Supreme Lord's opulence of knowledge. Narada represents the devotional service of the Supreme Lord. Devotional service is also represented by Lord Caitanya, who is considered to be the full representation of devotional service. In Brahma the opulence of creative power is invested, and in King Prthu the power for maintaining the living entities is invested. Similarly, in Parasurama the power for killing evil elements is invested.
Recognizing The Bonafide Avatara
When Lord Caitanya described the incarnation for this age of Kali, Sanatana Gosvami, who had been a government minister and was perfectly capable of drawing conclusions, directly asked of the Lord, "How can one understand the advent of an incarnation?" By the description of the incarnation for the Kali millennium, Sanatana Gosvami could understand that Lord Caitanya was indeed that incarnation of Krsna, and he could also understand that in the future there would be many people who would try to imitate Lord Caitanya.
"As one can understand the different incarnations for different millenniums by referring to Vedic literature," the Lord replied, "one can similarly understand who is actually the incarnation of Godhead in this age of Kali." In this way the Lord especially stressed reference to authoritative scriptures. In other words, one should not whimsically accept a person as an incarnation but should try to understand the characteristics of an incarnation by referring to scriptures. An incarnation of the Supreme Lord never declares Himself to be an incarnation, but His followers must ascertain who is an incarnation and who is a pretender by referring to authoritative scriptures. Real avatara does not want to be glorified (SB 4.15.23):
parokse 'smad-upasrutany alam
na stavayanti sabhyah
O gentle reciters, offer such prayers in due course of time, when the qualities of which you have spoken actually manifest themselves in me. The gentle who offer prayers to the Supreme Personality of Godhead do not attribute such qualities to a human being, who does not actually have them.
"Gentle devotees of the Supreme Personality of Godhead know perfectly well who is God and who is not. Nondevotee impersonalists, however, who have no idea what God is and who never offer prayers to the Supreme Personality of Godhead, are always interested in accepting a human being as God and offering such prayers to him. This is the difference between a devotee and a demon. Demons manufacture their own gods, or a demon himself claims to be God, following in the footsteps of Ravana and Hiranyakasipu. Although Prthu Maharaja was factually an incarnation of the Supreme Personality of Godhead, he rejected those praises because the qualities of the Supreme Person were not yet manifest in him. He wanted to stress that one who does not actually possess these qualities should not try to engage his followers and devotees in offering him glory for them, even though these qualities might be manifest in the future. If a man who does not factually possess the attributes of a great personality engages his followers in praising him with the expectation that such attributes will develop in the future, that sort of praise is actually an insult."
Any intelligent person can understand the characteristics of an avatara by understanding two features - the principal feature, called personality, and the marginal features. In the scriptures there are descriptions of the characteristics of the body and the activities of an incarnation, and the description of the body is the principal feature by which an incarnation can be identified. The activities of the incarnation are the marginal features. This is confirmed in the beginning of Srimad-Bhagavatam (1.1.1) where the features of an avatara are nicely described. In that verse, the two terms param and satyam are used, and Lord Caitanya indicates that these words reveal Krsna's principal feature. The other marginal features indicate that He taught Vedic knowledge to Brahma and incarnated as the purusa-avatara to create the cosmic manifestation. These are occasional features manifest for some special purposes. One should be able to understand and distinguish the principal and marginal features of an avatara. No one can declare himself an incarnation without referring to these two features. An intelligent man will not accept anyone as an avatara without studying the principal and marginal features. When Sanatana Gosvami tried to confirm Lord Caitanya's personal characteristics as being those of the incarnation of this age, Lord Caitanya Himself indirectly made the confirmation by simply saying, "Let us leave aside all these discussions and continue with a description of the saktyavesa-avataras."