In Assassin's Creed Brotherhood, the fictional device that lets Desmond relive Ezio's memories is the Animus.
The Animus is a machine developed by Abstergo that reads genetic memories encoded in a person's DNA. Desmond Miles uses it to experience the memories of his ancestor Ezio Auditore, allowing the modern-day story to investigate historical events and search for information connected to the Assassin-Templar conflict.
Brotherhood continues the Animus concept established in the first Assassin's Creed. Its historical sequences take place mainly in Renaissance Rome, while Desmond and his allies operate in the modern day. The device provides the narrative link between those two timelines.
Several related terms are easy to mix up. The Bleeding Effect is a side effect of prolonged Animus exposure, not the machine itself. An Apple of Eden is a powerful Isu artifact, and the Observatory is a location and information system featured elsewhere in the series. The Animus is the technology that makes memory reliving possible.