A couple of others:
www.redhotpawn.com
Here, you can play chess against players from all over the world. This is mostly a "correspondence" chess site, so you can set a game so that each player makes one move per day, or have longer time controls if you and your opponent agree. You can play in real time if you and your opponent have access to the site at the same time, but that's something you'd have to arrange.
Basically, on this site, you make your move, your opponent is notified by email that it is now their turn to move, and then when they make their move, you receive an email notifying you that it's your turn to move.
The site also has discussion forums on chess, as well as just about every other subject you can think of... sports, politics, religion, science/math, etc.
The site does keep track of your wins and losses, and you can access and review any of your past games (or games in progress) at any time. The good thing about this site is that you can have several games going at once, against different players. It also starts you out with a provisional rating, and adjusts your rating accordingly depending upon your results.
Another is:
http://www.chessworld.net/chessclubs/asplogin.asp
Pretty much the same deal here. It's a correspondence site.
It sounds like you might actually be looking for sites where you can play a full game in real time, rather than over several days or weeks. But if you are interested in correspondence chess, the two sites I listed are very good.
btw - each site allows "non-subscribers" to play a limited number of simultaneous correspondence games, but if you're a subscriber, the number of games that you can be playing at once is unlimited.
You might want to check these out. It's cool to have a lot of games going, it'll give you something to always think about.