A horcrux is an object created using dark magic to attain effective immortality. The concept is first introduced in the sixth novel, Harry Potter and the Half-Blood Prince, although horcruxes are present in earlier novels without being described or expanded upon.
A horcrux is created when a soul-shard split from a murderer's soul is infused into an object, which is then hidden or in some other manner kept safe. (When a person commits murder, his soul becomes traumatized and splits into more than one piece. A horcrux is nothing more than the fused object/soul-shard created when the portion of the murderer's soul that was split off is infused into some material object.) The point of creating a horcrux is to prevent the passage of a soul to the afterlife (death) by anchoring a portion of the soul in the material world.
Ordinarily, when one's body is killed, the soul departs for the next world. If, however, the body of a horcrux owner is killed, that portion of his soul which had remained in his body will not pass on to the next world, but will rather exist in a non-corporeal form capable of being resurrected by another wizard, as in The Goblet of Fire. If all of someone's horcruxes are destroyed, then his soul's only anchor in the material world would be his body, the destruction of which would then cause his final death.
Read more about this topic: Magic In Harry Potter, Dark Arts