WHO OR WHAT REINCARNATES?

According to Vedanta, the jiva reincarnates.  The jiva is the Atman identified with its coverings--the body, mind, senses, etc.  Ignorant of its divinity, the jiva experiences birth, death, pleasure and pain.
 
Meanwhile, Vedanta adopted the concept of a “subtle body” which is attached to the jiva for as long as the jiva’s bondage to samsara lasts.  Samsara is the perpetual cycle of birth, death, and rebirth.  The subtle body is what carries karmic debts.  It cannot, however, preserve one’s personal attributes.

The facts recorded by the subtle body are a sum of hidden tendencies or impressions imprinted by karma as seeds that will generate future behavior and personal character.  These tendencies will materialize unconsciously in the reborn individual, denying the person, at the same time, any hint of what his or her karmic condition actually is.
 
No form of transmitting conscious memory from one life to another is possible, as memory belongs to the world of illusion and dissolves at death.

As long as he remains unaware of his identity with Brahman, the jiva is reborn as a new individual with a new subtle body.  When at last the jiva realizes his true identity he awakens, freeing himself from the trap of samsara.