The avatar doctrine holds that when order in the world decays, Vishnu descends into it in a form suited to the emergency. The word means a crossing down. It is the mechanism that makes Vishnu the most historically absorbent of the great Hindu gods: a figure worshipped in one region under one name can be received as a form Vishnu took, without anyone being told their god was false.
The standard sequence is ten. Matsya the fish, Kurma the tortoise, Varaha the boar, Narasimha the man-lion, Vamana the dwarf, Parashurama the axe-bearer, Rama, Krishna, then the Buddha or Balarama, then Kalki, who has not yet come. The list is often presented as a progression from aquatic to amphibian to mammal to part-human to human, and read as an anticipation of evolution. It is not that, and the reading is a nineteenth- and twentieth-century one applied from outside; the sequence is a theological and narrative arrangement, and the texts show no interest in descent with modification.
The count is not fixed. Different Puranas give different numbers, and the Bhagavata Purana lists twenty-two. Membership varies too, and the ninth position is the clearest case: some lists give the Buddha, framed either as a compassionate deception or as a genuine descent, and some give Balarama instead. Where the Buddha is included, the inclusion functions as a claim over a rival tradition, and Buddhists have never accepted it.
Whether the avatars are appearances of one deity or absorbed local gods retrospectively identified with him is argued both ways, and the honest answer is that the doctrine works in both directions at once. It gave regional cults a route into a pan-Indian framework and gave the framework a way of growing without schism.