According to the Buddhist doctrine of samsara, what is reborn is a person's soul. The soul is perpetually reborn into other bodies until it finally (if it ever does) reaches nirvana and is released from this "wheel of life."
According to Buddhist doctrine, the soul moves from one body to another after the physical death of the first body. A soul can move into a variety of different bodies, both human and non-human, depending on the actions that it takes during each life. A soul that led a virtuous life will be reborn into a better state while one that was not virtuous and built up bad karma will be reborn into a lower state.
So, what is reborn in the cycle of samsara is the soul.
samsāra is typically described by Buddhists as a "Wheel of Suffering" or "Wheel of life." Entrapment within samsāra is conditioned by akushala, or, the three roots of suffering: dvesha (hatred), trishna (desire or craving) and avidya (delusion).
Buddhists originally accounted for the process of rebirth by appeal to phenomenological or psychological constituents.
Theravadins, for instance, identify consciousness as the link between death and rebirth. Although there is no existence of self, perpetual ignorance from moment to moment causes every changing psychological states (or the skandhas) to be perceived to be indicators of selfhood. As long as mental representations of self persist, so too does the cycle of rebirth. Theravada, therefore, places the realm of samsāra in direct opposition to nirvana, though the Mahayana and Vajrayana schools actually equate the two realms, considering them both to be devoid (or "empty") of essence. If everything is a mental representation, then so too are both samsāra and nirvana, which are nothing more than labels without substance. In these schools, realizing this simple fact allows for the realization that samsāra itself is the sole attainment, and existence is nothing other than the moment as it is.
Others schools of Buddhism dealt with the difficult coexistence of the samsāra and anatman doctrines in different ways. For instance, the Pudgalavāda school resurrected the concept of a "person" (pudgla) which transmigrates after death. Although this concept of a "person" is not necessarily equated with conceptions such as atman, such a teaching very nearly contradicts the notion of anatman. Another concept used by this school as well the Sarvastivadins to explain rebirth was that of antarabhava. This doctrine suggested the existence of an "intermediate being" present between life and rebirth. This being scouts out the location where rebirth is to occur as is dictated by karma from the previous life, and proceeds to attach itself to the sexual organs of the prospective parents of new child in which the soul will dwell.