Computer or A.I. characters in Sci-fi
o HAL-9000. One of the most famous.
o SAL-9000. His sister.
o The Master Control Program a.k.a. MCP in Tron.
o SKYNET from the Terminator movies, despite not having any screen time.
o The Puppetmaster a.k.a. Project 2501 in Ghost in the Shell.
o Colossus - from the eponymous movie.
o Mother and Father - from the Alien movie series.
o ORAC - smartass from Blake's 7.
o Eddie the Shipboard Computer - from Hitchhiker's Guide to the Galaxy.
o Dr. Theophilus - from the Buck Rogers in the 25th Century TV show (he wasn't mobile, he was a frisbee).
o Antikythera - From Dr. Who.
o Holly - from Red Dwarf.
o Joshua - in War Games.
o K.I.T.T. (and K.A.R.R.) from Knight Rider, although this is on the edge, as they were mobile. (thanks rootbeer77!)
o The Matrix.
o Melchior, Balthasar, and Caspar (spelling help from lj - thanks!) - from Neon Genesis Evangelion.
o Neuromancer and Wintermute from William Gibson's Neuromancer
o Continuity from Mona Lisa Overdrive by William Gibson (thanks Zerotime!)
o AIVAS from the Pern series by Anne McCaffrey (also from Zerotime)
o Frost, The Beta-Machine, Divcom and Solcom from Roger Zelazny's excellent For A Breath I Tarry.
o H.A.R.L.I.E. from When HARLIE was one
o VISAR, JEVEX and Zorac, the Shapieron's computer from James P. Hogan's Giants Trilogy.
o Cortana from Halo: the Fall of Reach and the game Halo (Thanks mirv!)
o Jane from the Ender trilogy.
o Aleph, Memex (aka HeyMex) and the Adviser (aka Mr Johnson) From Tom Maddox's superb novel Halo.
o Mandrake - Jack Butler's novel Nightshade.
o Mycroft Holmes - Heinlein's The Moon Is A Harsh Mistress.
o Minerva, Athene and Gay Deceiver - Time Enough For Love and its sequels.
o The Librarian - from Snow Crash.
o Ghostwheel - from the Amber series by Zelazny.
o AM - in Harlan Ellison's I Have No Mouth, And I Must Scream.
o The Central Computer ("CC") from John Varley's Steel Beach.
o Master System - from Jack L. Chalker's series, The Rings of the Master.
o Big Boppers - from Rudy Rucker's Software, offers eponymous.
o The First Sirian Bank from Terry Pratchett's book The Dark Side of the Sun, contributed by Zerotime.
o Melchizedek, the computer which runs Salem in the manga Gun Dream, a.k.a. Battle Angel Alita (from Glowing Fish).
o Hex from the Discworld series, also by Pratchett, contributed by Uberbanana.
o Deep Thought from Hitchhiker's Guide to the Galaxy, since as Uberbanana points out he's got a (critical) speaking part.
o Hactar from Life, the Universe and Everything, according to Oneiromancer and Polkovnik.
o The Solid State Entity from Neverness, by Zindell, contributed by auraseer.
* Games, RPG
o The Computer from Paranoia! (thanks again mirv!)
o SHODAN - System Shock.
o Xerxes - System Shock 2.
o Ava Johnson, Daedalus, Icarus, and Helios - AIs from Deus Ex.
o Durandal - he is the star in the Marathon FPS series for the Macintosh.
* Star Trek: TOS gets its own section. See also The Star Trek Project for more/better info!
o M-5 Duotronic - The Ultimate Computer (TOS)
o Landru - Return of the Archons (TOS)
o Losira - That Which Survives (TOS)
o Vaal - The Apple (TOS)
o The Oracle - For the World is Hollow, and I Have Touched the Sky (TOS)
o The Guardian Of Forever - The City on the Edge of Forever (TOS)