The Animus derives from the latin word for Mind. It is a futuristic looking machine that allows Desmond to live the past life of his ancestors. It is said that it homes in on certain parts of the DNA structure that tells and predicts your past life and allows to live the persons past life as if they were actually there. (Although in the game you control Altaïr from behind a 3rd person perspective).
It was created by the company Abstergo Industries and it's leading researcher(s), which we can only guess to be Dr. Warren Vidic. This machine is capable of locking onto genetic information and henceforth able to show the life to the subject visually. The Animus' job is to decode this genetic information and projects them in a visual manner inside the subjects head or 'Third Eye'. The effect the Animus has on people (or animals) cause unaccounted experience and intuition for the memory.
 In Game
In the game, the Animus serves as the game main screen UI where the player can continue with their campaign or load up a specific "memory" or level.
Although Altair is the focus of the game, the entirety of the story does not play out only in the world of Desmond's ancestors. In campaign mode as you complete various levels there are intermissions during which you control Desmond in the lab where the Animus is located. These are opportunities to learn more about Abstergo and Dr. Vidic and what the Animus is all about.