The Resurrection Stone is one of the sacred objects. Voldemort has been looking for the elder wand. Dumbledore thought Voldemort wanted to collect these three sacred objects, so he gave Harry the Resurrection Stone to stop Voldemort from collecting them. In fact, however, Voldemort just wants to get the elder wand to make himself more awesome, not to collect the Three Hallows.

Harry didn't come back to life at last. It was Harry who died as Voldemort's horcrux, not Harry himself. The Resurrection Stone can't bring people back to life, it can only bring people back to life, just like Harry Potter saw his parents and Sirius with the Resurrection Stone.

It was originally written that if the three sacred objects, the Elder Wand, the Resurrection Stone and the Invisibility Cloak, were put together and became their successors, they could dominate life and death.