In Assassin's Creed II, Ezio uses the Apple of Eden to unlock the Vault.
The Apple is one of the Pieces of Eden, powerful artifacts created by the Isu, the advanced civilization that existed before humanity's recorded history in the series' mythology. Its abilities include influencing minds and projecting images, making it valuable to both Assassins and Templars.
Ezio obtains the Apple during his campaign against the Templar conspiracy and carries it to the Vault beneath the Vatican. There, he encounters the holographic message of Minerva, whose warning connects the Renaissance story to the modern-day narrative.
The Staff of Eden is another Piece of Eden associated with Rodrigo Borgia and the Vatican sequence, but it does not provide the requested answer. The Vault requires the Apple, while the Staff plays a separate role in the finale.