Reincarnation

is the process by which the chakra of a deceased individual is reborn into a new, living vessel. It is also a term that refers to a person that is a result of this process.

History
After their deaths, the chakras of Asura and Indra — the two sons of the Sage of the Six Paths — would continue to be reincarnated in their descendants for centuries afterwards. Eventually, these chakras would be reborn in Hashirama Senju and Madara Uchiha respectively, who continued the brothers' struggle until their own deaths. Prior to his death, Madara, thanks to the machinations of Black Zetsu, became the first reincarnation to successfully mix the chakra of Asura and Indra, thus awakening the Rinnegan.

After Madara's death, the chakra of Asura and Indra continued the cycle of reincarnation, being reborn into Naruto Uzumaki — a descendant of Asura through the Uzumaki clan — and Sasuke Uchiha respectively. It also appears the reincarnation cycle of Indra and Asura doesn't start again until both of their current reincarnations have died, as Asura didn't reincarnate until decades after Hashirama's death. Shortly after Madara's second death, when both reincarnations were dead, both brothers reincarnated once again.

It also appears the chakra of Indra and Asura seem to still have a different chakra signature each time they change hosts as Madara was able to sense and distinguish Hashirama's chakra from Naruto even though he and Naruto both have Asura's chakra.

Known Reincarnations
Asura
 * Hashirama Senju
 * Naruto Uzumaki

Indra
 * Madara Uchiha
 * Sasuke Uchiha

Similar Cases

 * Through the Summoning: Impure World Reincarnation technique, both "reincarnation" and "resurrection" can be found; the sacrificed body is reincarnated into the body of the summoned shinobi, while the reincarnated shinobi is resurrected from the dead.
 * Reincarnation Ninjutsu are techniques that usually require or accomplish the transfer of life force between people.