The Templar artifact sought by Abstergo in the first Assassin's Creed is the Apple of Eden.
Within the game's mythology, the Apple of Eden is one of the Pieces of Eden, powerful artifacts created by the Isu, an advanced civilization that existed before recorded human history. The Apple can influence perception and behavior, making it valuable to the Templars, who seek control through order and centralized power.
Altaïr encounters the artifact during his investigation of the Templar conspiracy in the Holy Land. The Apple is initially hidden by Robert de Sablé and later becomes central to the game's final revelations. In the modern storyline, Abstergo wants access to the artifact's power and uses Desmond Miles's genetic memories to locate it.
The Apple is not the same as the Staff of Hermes, the Sword of Eden, or the Shroud of Eden. Those are separate Pieces of Eden associated with different characters and stories across the franchise. The first game's specific artifact is usually called the Apple of Eden, or simply the Apple.