The fictional organization that opposes the Assassins throughout the Assassin's Creed series is the Templars.
The Assassin-Templar conflict is the franchise's central fictional struggle. The Assassins generally defend free will and resist attempts to impose absolute control, while the Templars seek order through authority and powerful artifacts. Individual characters and methods vary, but the opposing philosophies recur across historical eras and modern-day storylines.
The Hidden Ones are the organization founded in Assassin's Creed Origins that later develops into the Assassin Brotherhood. The Order of the Ancients is the organization opposed by Bayek and Aya in Origins and is a predecessor to the Templars. The Initiates are a separate modern-day group and not the series' main opposing order.
The Templars draw on the historical Knights Templar as inspiration, but the game's organization is fictionalized and linked to the series' Isu artifacts and centuries-spanning conspiracy. This distinction explains why the answer is “the Templars,” rather than the name of a historical medieval order alone.