REINCARNATION IN BUDDHISM

There is no permanent self that reincarnates from one life to the next, Buddhism teaches, as compared to Vedanta where there is such a self, the Atman.  

Buddhism acknowledges, however, that something must reincarnate, in light of the Law of Karma.  That something is, in fact, karma.

Only karma passes from one life to another, since a person is merely the five skandhas comprising him, body, consciousness, sensations, cognition, and mental constructions that initiate actions, which disperse when the person dies.
 
Reincarnation where there is no transfer of a self was likened by the Buddha to the flame of a candle passed from another candle.  It is the same flame but different candles.
  
There is, however, no psycho-mental element transmitted between lives, the reborn person having no memory of his previous life, or of any of his past lives.

Buddhism underscores, meanwhile, the extreme rarity of human birth, a slim coincidence, indeed, when it happens, and an even slimmer coincidence should it be a Tathagata, a Buddha.

A person ceases to be reincarnated when all his karma has been worked through, and he experiences Nirvana.