Euro Game: A recent style in board gaming, first becoming widely popular with the mid-1990s success of The Settlers of Catan.
War Games: Board games that attempt to simulate realistic military conflict. Many are infamous for having Loads and Loads of Rules (and cardboard counters). Advanced Squad Leader is probably the most famous example.
Video Games: The most recent variety. Video games throw you into enthralling or addictive worlds where you forget yourself for a brief period. Any of the above categories can be translated to this format. Originating with Arcade Games, subgenres include Text, Adventure, MUDs, MMORPGs, RTS, FPS, and more.