What is Augmented Reality?
Have a look at this Wikipedia article. It does a much better job of explaining AR than we ever could.
What is the difference between Augmented Reality and Virtual Reality?
AR overlays virtual objects on top of a real world view. VR creates a completely fictitious world.
What is the difference between Augmented Reality and Mixed Reality?
This cuts a relatively fine line. In the strictest sense, Augmented Reality displays information on top of a real display. While Mixed Reality also overlays information on top of a real display, it enables you to interact with the virtual content.