According to Swami Parmeshwaranand, although the avatars of Vishnu are countless in number and include hermits, Manus, sons of Manus, and other Devas (Hindu Deity), due to a curse by the Rishi Bhrigu, most are only partial (i.e. incomplete) incarnations. The Dashavatara is a list of the ten complete (i.e., full) incarnations.
Various versions of the list of Vishnu's avatars exist, varying by region and tradition. Some lists mention Krishna as the eighth avatar and the Buddha as the ninth avatar, while others, such as the Paripāṭal (c. 3rd-4th CE), which is the fifth of the Eight Anthologies (Ettuthokai) in the Sangam literature, and the Yatindramatadipika, a 17th-century summary of Sri Vaishnava doctrine, list Balarama as the eighth avatar and Krishna as the ninth. The latter version is followed by some Vaishnavas who do not accept the Buddha as an incarnation of Vishnu. One list in the Mahabharata gives Rama (Bhargava), Rama (Dasharathi), Satvata (Krishna or Balarama), the Tri-Rama. The Tantric Prapanchasara (attributed to Adi Shankara, but disputed, ) also omits the Buddha. Though no list can be uncontroversially presented as standard, the "most accepted list found in Puranas and other texts is [...] Krishna, Buddha."

The following table summarises the position of avatars within the Dashavatara in many but not all traditions:
• Balarama-Samkarshana, holding the gada and plough, and Vasudeva-Krishna,holding the sankha (a conch) and chakra, the half-brother avatars of Vishnu, are the earliest representations of avatars. Mint of the non-Hindu coinage of Agathocles, an Indo-Greek ruler, circa 185-170 BCE
• Hindu god Vishnu's ten major avatars (Balarama-Krishna version) Dasavatara shrine, 18th century ivory (National Museum, New Delhi). From top descending: Matsya; Kurma; Varaha; Narasimha; Vamana; Parashurama; Rama; Balarama ; Krishna and Kalki.
• Hindu god Vishnu's ten major avatars (Balarama-Buddha version). From left: Matsya; Kurma; Varaha; Narasimha; Vamana; Parashurama; Rama; Balarama; Buddha; and Kalki
The Agni, Padma, Garuda, Linga, Narada, Skanda and Varaha Puranas mention the common (Krishna, Buddha) Dashavatara list. The Garuda Purana has two lists, one longer list with Krishna and Buddha, and a list with Balarama and Buddha, which substitutes Vamana for Rama. The Shiva Purana has Balarama and Krishna. The list with Krishna and Buddha is also found in the Garuda Purana Saroddhara, a commentary or 'extracted essence' of the Garuda Purana (i.e. not the Purana itself, with which it seems to be confused):

The Fish, the Tortoise, the Boar, the Man-Lion, the Dwarf, Parasurama, Rama, Krisna, Buddha, and also Kalki: These t…

Read more on Wikipedia

199 Matsya: The fish avatar. King Vaivasvata Manu finds a little fish in the palm of his hands when performing the tarpana (water-offering). The fish asks Manu if his riches and power was enough to give the fish a nice home. Manu keeps the fish to give it a home, but the fish keeps expanding, which breaks Manu's pride about his wealth. Eventually, he releases it into the ocean, realizing it is Vishnu himself. Vishnu informs Manu of the coming destruction of the world, by means of fires and floods, and directs Manu to collect "all creatures of the world" and keep them safe on a boat built by the gods. When the deluge (Pralaya) occurs, Vishnu appears as a great fish with a horn, to which Manu ties the boat, which leads them into safety.
299 Kurma: The tortoise/turtle avatar. In the legend of the Samudra Manthana, the devas and asuras were churning the Ocean of Milk in order to obtain amrita, the nectar of immortality. They used the mountain Mandara as the churning shaft, which started to sink. Vishnu took the form of a tortoise to bear the weight of the mountain to allow them to complete their task.
399 Varaha: The boar avatar. The gatekeepers of Vaikuntha, the abode of Vishnu, Jaya and Vijaya, are cursed by the Four Kumaras when they stop them from seeing Vishnu. They choose to be reborn three times as asuras as adversaries of Vishnu. In their first birth, they are born as the brothers Hiranyaksha and Hiranyakashipu. Varaha appeared to defeat Hiranyaksha, who had abducted the earth, and by extension the earth goddess, Bhumi, and carried it to the bottom of the cosmic ocean. The battle between Varaha and Hiranyaksha is believed to have lasted for a thousand years, which the former finally won. Varaha carried the earth out of the ocean between his tusks and restored it to its place in the universe.
499 Narasimha: The half-man/half-lion avatar. Hiranyakashipu persecuted everyone for their religious beliefs including his son, Prahlada, who was a devotee of Vishnu. The boy was protected by the god and could not be killed, thus being saved by the several attempts of getting harmed. Vishnu descended as an anthropomorphic incarnation, with the body of a man and head and claws of a lion. He disemboweled Hiranyakashipu, and brought an end to the persecution of human beings, including his devotee Prahlada.
599 Vamana: The dwarf avatar. The grandson of Prahlada, Bali, with devotion and penance was able to defeat Indra, the king of heaven. This humbled the other deities and extended his authority over the three worlds. The gods appealed to Vishnu for protection and he descended as a boy Vamana. During a yajna of the king, Vamana approached him and Bali promised him for whatever he asked. Vamana asked for three paces of land. Bali agreed, and the dwarf then changed his size to that of the giant Trivikrama form. With his first stride he covered the earthly realm, with the second he covered the heavenly realm thereby symbolically covering the abode of all living beings. He then took the third stride for the …

Read more on Wikipedia

The Buddha was included as one of the avatars of Vishnu under Bhagavatism by the Gupta period between 330 and 550 CE. The mythologies of the Buddha in the Theravada tradition and of Vishnu in Hinduism share a number of structural and substantial similarities. For example, states Indologist John Holt, the Theravada cosmogony and cosmology states the Buddha covered 6,800,000 yojanas in three strides, including earth to heaven and then placed his right foot over Yugandhara – a legend that parallels that of the Vamana avatar in Hinduism. Similarly, the Buddha is claimed in the Theravada mythology to have been born when dharma is in decline, so as to preserve and uphold the dharma. These similarities may have contributed to the assimilation of the Buddha as an avatar of Vishnu.

The adoption of Buddha as an avatar in Bhagavatism was a catalyzing factor in Buddhism's assimilation into Vaishnavism's mythic hierarchy. By the 8th century CE, the Buddha was included as an avatar of Vishnu in several Puranas. This assimilation is indicative of the Hindu ambivalence toward the Buddha and Buddhism, and there is also a tradition that there were two Buddhas. According to this tradition, the first was the ninth avatar of Vishnu, while the second was the historical Buddha. Conversely, Vishnu has also been assimilated into Sinhalese Buddhist culture, and Mahayana Buddhism is sometimes called Buddha-Bhagavatism. By this period, the concept of Dashavatara was fully developed.
Jayadeva, in his Pralaya Payodhi Jale from the Gita Govinda, includes Balarama and Buddha where Krishna is equated with Vishnu and the source of all avatars. The words from the poem describing ten avataras (translated) are as follows:

As a reviver of Vedas as a fish, bearer of this earth as tortoise, uplifter and supporter of earth as wild boar, slasher of Hiranyakashyapa as lion man, deluder of Bali as dwarf boy, annihilator of Kshatriya s as Parashu Rama, conqueror of Ravana, the legatee of Paulastya, as Rama, wielder of plough as bala raama, fosterer of non violence as Buddha, mangler of fractious races as Kalki, you alone can put on ten semblances, thus oh, Krishna, my reverences are unto you.

In traditions that emphasize the Bhagavata Purana, Krishna is the original Supreme Personality of Godhead, from whom everything else emanates. Gaudiya Vaishnavas worship Krishna as Svayam Bhagavan, or source of the incarnations. The Vallabha Sampradaya and Nimbarka Sampradaya, (philosophical schools) go even further, worshiping Krishna not only as the source of other incarnations, but also Vishnu himself, related to descriptions in the Bhagavata Purana. The Mahanubhavas, also known as the Jai Kishani Panth, consid…

Read more on Wikipedia