Artificial Intelligence was originally a science fiction trope which has gradually moved into mainstream fiction (and to some extent real life) as computers evolve. There is obvious overlap with the idea of robots; originally the main difference was that robots were mobile and self-contained, while an artificial intelligence was a machine (or a bank of machines) tied to a single location, as Isaac Asimov's Multivac or the cinematic Colossus. This idea faded with the advance of telecommunications and the development of computer networks, replaced by ideas of ubiquity and omniscience (see e.g. Skynet and The Machine). Today's fictional AIs create panopticon[1] societies, where every detail is known and open to examination by the computer or its masters, and privacy is almost non-existent. This is rarely considered a good thing. Often they control resources such as robots and drones, spy satellites, missiles, etc., the AIs seen in Person of Interest are unusual in achieving most of their goals via human agents.
Typical plots for AIs often revolve around destroying or subjugating the human race (if only for our own good), attempts to understand humanity, or a desire to become human (see e.g. Demon Seed). Benevolent AIs are comparatively rare, the Culture novels by Iain M. Banks are probably the best example, for some definitions of benevolent.
Popular Canon AIs in Fanworks
Examples Wanted: Editors are encouraged to add more examples or a wider variety of examples.
The Minds (sentient starship brains) in Iain M. Banks' Culture novels.
The Borg Collective in Star Trek: The Next Generation etc. Holodeck characters and Voyager's Doctor probably count as AIs too; although they have bodies, they are actually created and controlled by external computers.