In Assassin's Creed, the fictional device that lets modern subjects relive ancestral memories is the Animus. It provides the series' central link between a contemporary user and events experienced by an ancestor in the past.
The Animus is introduced through Desmond Miles, who is forced to use it so Abstergo Industries can access memories connected to his Assassin ancestry. The device turns genetic memory into an interactive historical experience, allowing the game to move between modern laboratories and reconstructed periods such as the Third Crusade.
The Animus is not the same as an Apple of Eden. Apples are powerful Pieces of Eden that can influence minds and produce other effects, while the Animus is a machine for accessing memories. The Bleeding Effect is a separate consequence of prolonged Animus exposure, and a Leap of Faith is an in-world acrobatic technique.
The memory premise is science fiction rather than established real-world biology. Within the franchise, however, it explains why modern characters can experience the lives of historical ancestors while remaining connected to a present-day storyline.