I haven't seen the film but this is my understanding from reading reviews, seeing the film's trailers and playing the games:
The film's story and characters are totally original - the setting (Medieval Spain) or characters don't appear anywhere in the games. However, I believe the main thrust of the in-game story arc - the timeless conflict between the Brotherhood of Assassins and the Templars - is portrayed in the film.
As I haven't seen the film, I don't know if it references anything in the games to say if it's a shared universe or not.
The animus technology is portrayed totally differently. In the games, it's just a bed with something over it (the design varies from game to game, but if you think of an MRI scanner you won't be far off). In the film it's kind of a robotic arm, and I understand the movements the character makes in the past are reflected in the present - i.e. jumping around the room. In the games, the present day character is sedate and unconscious when they are in the animus.
Bit more information here.