In the Siddhārtha-saṁhitā there is a description of the twenty-four forms of Viṣṇu, and these forms are named according to the position of the symbols in Their four hands. When describing the positions of objects in the hands of the Viṣṇu mūrtis, one should begin with the lower right hand and then move to the upper right hand, to the upper left hand and finally to the lower left hand. In this way, Vāsudeva is represented by club, conch shell, disc and lotus flower. Saṅkarṣaṇa is represented by club, conch shell, lotus flower and disc. Similarly, Pradyumna is represented by disc, conch shell, club and lotus flower. Aniruddha is represented by disc, club, conch shell and lotus flower. In the spiritual sky the representations of Nārāyaṇa are twenty in number and are described as follows: Śrī Keśava (lotus, conch shell, disc and club), Nārāyaṇa (conch, lotus, club and disc), Śrī Mādhava (club, disc, conch and lotus), Śrī Govinda (disc, club, lotus and conch), Viṣṇu-mūrti (club, lotus, conch and disc), Madhusūdana (disc, conch, lotus and club), Trivikrama (lotus, club, disc and conch), Śrī Vāmana (conch, disc, club and lotus), Śrīdhara (lotus, disc, club and conch), Hṛṣīkeśa (club, disc, lotus and conch), Padmanābha (conch, lotus, disc and club), Dāmodara (lotus, disc, club and conch), Puruṣottama (disc, lotus, conch and club), Acyuta (club, lotus, disc and conch), Nṛsiṁha (disc, lotus, club and conch), Janārdana (lotus, disc, conch and club), Śrī Hari (conch, disc, lotus and club), Śrī Kṛṣṇa (conch, club, lotus and disc), Adhokṣaja (lotus, club, conch and disc), and Upendra (conch, club, disc and lotus).
According to the Hayaśīrṣa-pañcarātra, there are sixteen forms, and these are also named according to the positions of the disc and so on. The conclusion is that the Supreme Original Personality of Godhead is Kṛṣṇa. He is called līlā-puruṣottama, and He resides principally in Vṛndāvana as the son of Nanda. It is also learned from the Hayaśīrṣa-pañcarātra that there are nine forms protecting the two Purīs known as Mathurā Purī and Dvārakā Purī. These nine forms are Vāsudeva, Saṅkarṣaṇa, Pradyumna, Aniruddha, Nārāyaṇa, Nṛsiṁha, Hayagrīva, Varāha and Brahmā. These are different manifestations of the prakāśa and vilāsa forms of Lord Kṛṣṇa.
Lord Caitanya next informed Sanātana Gosvāmī that there are different forms of svāṁśa as well, and these are divided into the Saṅkarṣaṇa division and the incarnation division. The Saṅkarṣaṇa division includes the three puruṣa-avatāras—Kāraṇodakaśāyī Viṣṇu, Garbhodakaśāyī Viṣṇu and Kṣīrodakaśāyī Viṣṇu—and the other division comprises the līlā-avatāras, such as the Lord's incarnations as a fish and a tortoise.
There are six kinds of incarnations: (1) the puruṣa-avatāras, (2) the līlā-avatāras, (3) the guṇa-avatāras, (4) the manvantara-avatāras, (5) the yuga-avatāras and (6) the śaktyāveśa-avatāras. Out of the six vilāsa manifestations of Kṛṣṇa, there are two divisions based on His age, and these are called bālya and paugaṇḍa. As the son of Nanda Mahārāja, Kṛṣṇa in His original form enjoys both of these childhood aspects—namely bālya and paugaṇḍa.
We can conclude that there is no end to the expansions and incarnations of Kṛṣṇa. Lord Caitanya described some of them to Sanātana just to give him an idea of how the Lord expands and how He enjoys. These conclusions are confirmed in Śrīmad-Bhāgavatam (1.3.26). There it is said that there is no limit to the incarnations of the Supreme Lord, just as there is no limit to the waves of the ocean.
Kṛṣṇa first incarnates as the three puruṣa-avatāras, namely the Mahā-viṣṇu or Kāraṇodakaśāyīavatāra, the Garbhodakaśāyīavatāra and the Kṣīrodakaśāyīavatāra. This is confirmed in the Sātvata-tantra. Kṛṣṇa's energies can also be divided into three: His energy of thinking, His energy of feeling, and His energy of acting. When He exhibits His thinking energy He is the Supreme Lord, when He exhibits His feeling energy He is Lord Vāsudeva, and when He exhibits His acting energy He is Saṅkarṣaṇa Balarāma. Without the Lord's thinking, feeling and acting, there would be no possibility of creation. Although there is no creation in the spiritual world as there is in the material world, both worlds are manifestations of Kṛṣṇa's energy of acting, which He carries out in the form of Saṅkarṣaṇa Balarāma.
The spiritual world—the Vaikuṇṭha planets and Kṛṣṇaloka—is situated in Kṛṣṇa's energy of thinking. Although there is no creation in the spiritual world, which is eternal, it is still to be understood that the spiritual planets depend on the thinking energy of the Supreme Lord. This thinking energy is described in the Brahma-saṁhitā (5.2), where it is said, "The supreme abode, known as Goloka, is manifested like a lotus flower with hundreds of petals. Everything there is manifested by Ananta, who is a form of Balarāma, or Saṅkarṣaṇa." The material cosmic manifestation and its different universes are manifested through māyā, or the material energy, but one should not think that the material energy, or material nature, is the cause of this cosmic manifestation. Rather, it is caused by the Supreme Lord, who uses His different expansions to act through material nature. In other words, there is no possibility of any creation without the superintendence of the Supreme Lord. The form of the Lord who causes the energy of material nature to bring about creation is Saṅkarṣaṇa, and it is understood that this cosmic manifestation is created when the material nature comes in contact with the superintendent energy of the Supreme Lord, Saṅkarṣaṇa. The example is given of iron becoming hot in contact with fire and, when red hot, acting just like fire.
In Śrīmad-Bhāgavatam (10.46.31) it is said that Balarāma and Kṛṣṇa are the origin of all living entities and that these two personalities enter into everything. A list of incarnations is given in Śrīmad-Bhāgavatam (1.3), and they are as follows: (1) the Kumāras, (2) Nārada, (3) Varāha, (4) Matsya, (5) Yajña, (6) Nara-Nārāyaṇa, (7) Kārdami Kapila, (8) Dattātreya, (9) Hayaśīrṣa, (10) Haṁsa, (11) Dhruvapriya, or Pṛśnigarbha, (12) Ṛṣabha, (13) Pṛthu, (14) Nṛsiṁha, (15) Kūrma, (16) Dhanvantari, (17) Mohinī, (18) Vāmana, (19) Bhārgava (Paraśurāma), (20) Rāghavendra, (21) Vyāsa, (22) Pralambāri Balarāma, (23) Kṛṣṇa, (24) Buddha and (25) Kalki. Because almost all of these twenty-five līlā-avatāras appear in one day of Brahmā, which is called a kalpa, they are sometimes called kalpa-avatāras. Out of these incarnations, Haṁsa and Mohinī are not permanent, but Kapila, Dattātreya, Ṛṣabha, Dhanvantari and Vyāsa are five eternal forms, and they are more celebrated. Kūrma (the tortoise incarnation), Matsya (the fish), Nara-Nārāyaṇa, Varāha (the boar), Hayaśīrṣa, Pṛśnigarbha and Balarāma are considered incarnations of vaibhava. Similarly, there are three guṇa-avatāras, or incarnations of the qualitative modes of nature, and these are Brahmā, Viṣṇu and Śiva.
There are fourteen manvantara-avatāras: (1) Yajña, (2) Vibhu, (3) Satyasena, (4) Hari, (5) Vaikuṇṭha, (6) Ajita, (7) Vāmana, (8) Sārvabhauma, (9) Ṛṣabha, (10) Viṣvaksena, (11) Dharmasetu, (12) Sudhāmā, (13) Yogeśvara and (14) Bṛhadbhānu. Out of these fourteen manvantara-avatāras, Yajña and Vāmana are also līlā-avatāras. The manvantara-avatāras are also known as vaibhava-avatāras.
The four yuga-avatāras are also described in Śrīmad-Bhāgavatam. In the Satya-yuga the incarnation of God is white; in the Tretā-yuga He is red; in the Dvāpara-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 Mahāprabhu). As far as the śaktyāveśa-avatāras are concerned, they include Kapila, Ṛṣabha, Ananta, Brahmā (although sometimes the Lord Himself becomes Brahmā), Catuḥsana (the Kumāras, who are the incarnation of knowledge), Nārada (the incarnation of devotional service), King Pṛthu (the incarnation of administrative power) and Paraśurāma (the incarnation who subdues evil principles).