The precursor race whose technology appears throughout Assassin's Creed is called the Isu. In the series' modern terminology, they are also known as the First Civilization or Those Who Came Before.
The Isu were an advanced civilization that existed before recorded human history. They created powerful Pieces of Eden, including the Apple of Eden, and used them to control or influence early humans. Their civilization was devastated by a solar catastrophe, after which humans inherited a world shaped by Isu artifacts and surviving genetic memories.
The Assassins and Templars are human organizations that debate how society should be governed; they are not the precursor race. “First Civilization” is a valid alternative label for the Isu, but “The Isu” is the series' distinctive canonical name and the concise answer requested here.