In Assassin's Creed II, the device that lets Desmond experience ancestral memories is the Animus.
The Animus is a machine developed by Abstergo to access genetic memories inherited through a person's ancestry. Desmond uses it to relive Ezio Auditore's life, while the player experiences those memories as the game's historical missions.
The concept allows the series to move between a modern storyline and different historical periods without treating time travel as the mechanism. The Animus also supplies the fictional explanation for synchronizing with an ancestor and restarting after certain failures.
The Apple of Eden is a separate Isu artifact, not a memory device. The Bleeding Effect is the mental transfer of skills and traits caused by prolonged Animus exposure, so it describes a consequence rather than the machine itself.