Rebirth in
Buddhism is distinctly different from
Reincarnation or
Transmigration of an
immortal soul.
For the Buddha, there was no soul to move from one body to the next, and so the idea of such a soul migrating in such a fashion was a logical impossibility.
Instead, Rebirth in the Buddhist sense of things can be looked at in the following way: we go through our lives, each moment passing by. The person who we are now is affected by the karma (kamma) we have generated in this and previous lives. Furthermore, the person we are now is different than the person we were, say, ten years ago, but there is something which binds those two beings together.
Much in the same way, as we are the same being we were in the last moment, and the same being we shall be in the next, so too are we the same being we were in the last life, the same being we shall be in the next.