yeah, there are a lot of different kinds of games to play online.
online/multiplayer chess, checkers, card games, pool and such can be readily found at places like Yahoo! Games for free.
http://games.yahoo.com/
then there are retail games, which usually come in 2 varieties..... the kind you pay for once at a premium (Quake, Counter Strike, Unreal Tournament and others to name a few widely known ones), and the kind you pay a monthly subscription for (as well as a smaller price for the game itself in the store), like World of Warcraft, Star Wars Galaxies, Everquest and so on. typically, these are MMORPGs (massive multiplayer online role playing games..... more emphasis on role playing instead of realtime first-person combat like with Quake and CS. though there are RPG games that are not subscription-based.
the game field is actually much bigger and more diverse than i am making it sound, but i am trying to keep it in a nutshell.