I enjoy computer games, but I don't play them very often.
The first computer game I played definitely remember was Star Trek on a PDP-11 running RSTS, but I may have played one on a PDP-8 before that.
The last computer game I played with any frequency was Diablo II, but I haven't played it much since I lost regular access to multi-player games with a bunch of good friends.
Some games that I've enjoyed:
Classic Interactive Fiction: Colossal Cave, Adventure, Zork.
First Person Shooters: Apple II+ Castle Wolfenstien, Doom.
Third Person Shooters: Diablo II.
Virtual Worlds: Myst.
Classic Strategy/Exploration: RSTS/E Star Trek, VAX Empire.[1]
[1] Played on the DEC GIGI graphics terminal: love that crossed diamond cursor.