Reincarnation is considered, in Hinduism, to simply be part of the universal process, which includes life, death and rebirth, and is governed by karma. Every action produces a karmic result which must be lived out at some point in this lifetime or a future lifetime. It is the attachment to material desires and ignorance which perpetuates the cycle of rebirth. This cycle continues until moksha, or "liberation," is achieved.
Within Buddhism, the concept of rebirth is differentiated from other teachings on reincarnation in that there is no individual soul or spirit which lives on, but, instead, it is merely consciousness which connects one life with another. This fits with what Buddhist teachers of meditation say is observed once the attention is turned inward - that there is no continuum of awareness, but instead life is experienced one moment of awareness at a time. The transition from one life to another is simply the continuation of this moment-to-moment shift of consciousness.