The artifact most commonly called the Apple of Eden is a Piece of Eden. In Assassin's Creed lore, Pieces of Eden are advanced artifacts created by the Isu, an ancient civilization that existed before humanity's recorded history.
The Apples are spherical Pieces of Eden associated with powerful effects such as controlling minds, creating illusions, and manipulating perception. Their abilities vary by story and are presented through the series' science-fiction framework rather than as ordinary magic.
Both Assassins and Templars seek these artifacts. The Templars generally want to use them to impose order and control, while the Assassins oppose centralized domination and try to keep such power from being abused.
The term Apple of Eden describes one type of Piece of Eden, not every artifact in the category. Other examples include the Staff of Hermes Trismegistus, the Sword of Eden, and the Shroud of Eden. The Animus is different: it is a machine used to access genetic memories.