The device that lets Assassin's Creed users relive genetic memories is called the Animus.
Within the series, the Animus is a machine that interprets genetic memories encoded in a person's DNA. A modern subject enters the device and experiences the memories of an ancestor, allowing the narrative to move between a present-day storyline and historical settings.
Desmond Miles is the best-known early Animus subject. Through him, players experience the memories of Altaïr, Ezio, and Ratonhnhaké:ton. Later games use different users and versions of the technology, including Abstergo's Helix system.
The Animus is fictional and should not be confused with real genetic memory. Real DNA can preserve biological information, but science does not support the idea that detailed personal experiences can be retrieved from it. Eagle Vision is also different: it is an Assassin's Creed perception ability, not a machine.