There are many definitions of the reincarnation concept. The main one is a soul rebirth in a new body.
The first definition of reincarnation came from Pythagoras. He was a Greek philosopher and mathematician. He taught that the soul was immortal and merely resides in the body. Therefore, it survived bodily death. His further teachings held the soul goes through a series of rebirths. Between death and rebirth the soul rests and is purified in the Underworld. After the soul has completed this series of rebirths it becomes so purified that it can leave the transmigration or reincarnation cycle.
Another definition from Plato. Another Greek philosopher. He shared similar views as Pythagoras. The soul of man is eternal, pre-existence, and wholly spiritual.
In Plato’s view of the transmigration of the soul from body to body. However, there is a difference. Plato claimed the soul tends to become impure during these bodily inhabitations although a minimal former life knowledge remains. However, if through its transmigrations the soul continues doing good and eliminates the bodily impurities it will eventually return to its pre-existence state.
But, if the soul continually deteriorates through its bodily inhabitations it will end up in Tartarus, a place of eternal damnation. This appears to be an origination of both the concept of karma and the Christian concept of hell.
Egyptians believed in reincarnation or the transmigration of the soul. They thought the soul transmigrated from body to body. And this was a reason why they embalmed the body in order to preserve it. This establishes the dating of the concept of reincarnation back to the ancient Egyptian religion but many think it dates beyond antiquity.
These are some related words and synonyms.
- Transmigration of the soul.
Here is an informative page on Reincarnation in Japanese-Buddhism.