Reincarnation

is a process through which the chakra of a deceased individual is reborn in a new living vessel. The living vessel is also referred to as a reincarnation.

Overview
For centuries after their deaths, the chakras of the sons of the Sage of the Six Paths, Indra and Asura Ōtsutsuki, would continue to be reincarnated into their descendants. During the Warring States Period, their chakras would be reborn into Madara Uchiha and Hashirama Senju, who would continue the brothers' struggle until their own deaths. Prior to Madara's death, Black Zetsu would manipulate him into becoming the first reincarnation to successfully merge Indra's and Asura's chakras, and this would lead to the awakening of the Rinnegan.

After Madara's death, the chakra of Indra and Asura would be reborn into Sasuke Uchiha of the Uchiha clan, and Naruto Uzumaki — a descendant of Asura through the Uzumaki clan, respectively. As the first reincarnations to have been born into an era of relative peace, the two would maintain a rather civil rivalry until Sasuke would fall victim to his clan's Curse of Hatred.

The reincarnation cycle will not start again until both of the current reincarnations have passed away; the brothers would not reincarnate until Madara's death, which would occur decades after Hashirama's.

The chakras of Indra and Asura change chakra signatures with each reincarnation. Although Hashirama and Naruto are both Asura's reincarnations, Madara, a sensor type, was able to differentiate their chakras. However, the chakra pathways do not appear to change, or at the very least, contain similarities. Whilst using the Byakugan, Kaguya Ōtsutsuki was able to note that Naruto and Sasuke were the reincarnations of her grandsons.

Known Reincarnations
Asura Ōtsutsuki
 * Hashirama Senju
 * Naruto Uzumaki

Indra Ōtsutsuki
 * 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.
 * Both Hagoromo Ōtsutsuki, the Sage of Six Paths, and his mother Kaguya Ōtsutsuki were spirtually manifested when Obito & Madara Uchiha achieve similar existences to the two. Following the absorption of the Ten-Tails and becoming its host, with the possession of a Rinnegan, Obito unknowingly willed the Sage of Six Paths back into existence, and thus allowed for the progenitor of shinobi to pass on the Six Paths Chakra to his reincarnated sons. Upon fulfilling the prophecy to sprout the Rinne Sharingan and cast the Infinite Tsukuyomi, Madara, as the host of the Ten-Tails, becomes a similar existence to Kaguya---however, only with the assistance of Black Zetsu could she be physically revived.
 * Obito Uchiha has shown the ability to transmigrate and enter living vessels to bestow them the power of his eyes, witnessed after his death, Obito's spirit returns from the Pure Land and possesses Kakashi Hatake to temporarily awaken the Mangekyō Sharingan in both of his eyes. In Naruto: Road to Ninja, Obito (as the "Masked Man") possesses the unconcious body of Menma Uzumaki, allowing him to fight inside the vessel with his Sharingan Kekkei Genkai.

Trivia

 * In the official subs for episode 421 of the Naruto: Shippūden, Hagoromo stated that the first generation of his sons' reincarnates was that of Madara and Hashirama; however, in the manga, it was stated there were numerous other reincarnates before them.