Reincarnation (or rebirth) is the claim that what one is doing now continues after physical death, taking up another form. In Hinduism and Jainism, it is the Atman or soul that passes. In Buddhism — which rejects a permanent self — it is a continuity of Karma without a substance, often compared to one candle lighting another.
The tradition usually distinguishes reincarnation from a view that the current person persists intact into the next life. What persists is something much more lightweight — tendencies, traces, the grammar of a mind-stream.
Whether this is literal or a compelling pedagogy for the weight of one’s choices is a question held differently within each tradition.