The fictional order opposing the Assassins throughout Assassin's Creed is the Templar Order, commonly called the Templars.
The Templars seek peace through control, while the Assassins defend freedom of choice and resist centralized domination. Their conflict stretches across different historical periods, with each game placing the rivalry in a new political and cultural setting.
The series uses related enemy organizations in some eras. The Order of the Ancients predates the formal Templar Order in Assassin's Creed Origins, and the Cult of Kosmos is a separate faction in Odyssey. The Borgia family also serves as an important enemy force, but it is not the name of the overarching opposing order.