The fictional ancient civilization that precedes humanity in Assassin's Creed lore is called the Isu.
The Isu are an advanced species also known as the Precursors or Those Who Came Before. In the series' mythology, they existed before modern humans and created humans as a labor force. Their civilization developed powerful artifacts, many of which later became known as Pieces of Eden.
The Isu-human conflict and the catastrophe that devastated their world form a major part of the franchise's overarching mythology. Human legends often preserve distorted memories of Isu individuals, places, and technologies, allowing the games to connect mythology with their fictional ancient history.
“First Civilization” is another label used for the Isu, so it is closely related rather than an unrelated rival group. The Templars and Hidden Ones are human organizations formed much later. Pieces of Eden are artifacts, not the civilization itself.