The artifact sought by Desmond Miles in Assassin's Creed is the Apple of Eden.
The Apple is one of several Pieces of Eden, technologically advanced artifacts created by the Isu, the powerful precursor civilization in the series' mythology. It can influence human minds, project illusions, and reveal information through visions. The Templars pursue it because its abilities could help control society.
In the first game, Altaïr Ibn-La'Ahad encounters the Apple after the Battle of Arsuf and carries it to Masyaf. In the modern-day storyline, Desmond experiences Altaïr's memories through the Animus, making the artifact central to both historical and contemporary plotlines. The Apple should not be confused with the Staff, Sword, or Shroud, which are separate Pieces of Eden with different powers.