Reincarnation

Choose target creature. When that creature dies this turn, return a creature card from its owner's graveyard to the battlefield under the control of that creature's owner.

"You cannot imagine what eyes you may look through tomorrow."
—Aerona, elder druid

Commander 2013

Masters Edition III

Choose target creature. When that creature is put into a graveyard this turn, return a creature card from that graveyard to the battlefield under the control of that creature's owner.

Legends

If target creature is placed in graveyard this turn, bring a creature from that graveyard directly into play under the control of the owner of the target creature. Treat this creature as though it were just summoned.

All Reincarnation does when it resolves is set up a delayed triggered ability. If the targeted creature isn't put into a graveyard this turn, that ability never triggers. If the targeted creature is put into a graveyard this turn, that ability triggers and resolves like any other triggered ability.

The creature card that will be returned to the battlefield isn't chosen until the delayed triggered ability resolves. The player who chooses is the player who controlled Reincarnation. The player under whose control the chosen creature card enters is the player who owned the targeted creature (in other words, the player into whose graveyard it went), not necessarily the controller of that creature.

The creature that's returned to the battlefield may be the same one that was put into the graveyard (assuming it's still in the graveyard when the delayed triggered ability resolves). If so, it returns to the battlefield as a brand-new creature.