The artifact sought by both factions throughout the original Assassin's Creed is the Apple of Eden.
The Apple is one of the Pieces of Eden, powerful artifacts created by the precursor civilization known as the Isu. In the first game, the Templars seek to use it to control human behavior, while the Assassins oppose that goal. Altaïr becomes involved in the struggle after his missions reveal the artifact's importance.
The Apple is often described as a crystal-like sphere, but its significance is more important than its appearance. It can produce illusions and influence minds, making it a symbol of the series' recurring conflict between control and free will.
Other Pieces of Eden appear across the franchise, including swords, staffs, and shrouds. Those names are plausible alternatives, but the original game's central artifact is specifically the Apple of Eden.