In the Assassin's Creed series, the Templars are the organization that traditionally opposes the Assassins.
The conflict is built around two competing visions of human society. The Assassins defend free will and resist imposed control, while the Templars seek peace and order through power, authority, and control. Their struggle extends across many historical periods and is presented through the series' fictional reinterpretation of history.
The modern Templar organization is associated with Abstergo Industries, a corporation that uses the Animus and related technology to explore genetic memories. Abstergo is therefore a modern institution connected to the Templars, not the name used for the historical faction in every era.
The Hidden Ones are the forerunners of the Assassin Brotherhood in Assassin's Creed Origins. The Medjay are an ancient Egyptian protective order associated with Bayek. Neither is the series-wide opposing organization asked for here.