There are a lot of different Chess sites on the internet. Some of them enable you to play chess, some to learn chess, some are about chess news and some are dealing with the chess adventures of the author.
Chess.com is the major Chess site and it is dealing with all of these aspects of chess.
On Chess.com you can play both turn-based correspondence-style chess as well as live chess and even tournaments.
I also like the Chess Mentor, but these Chess Lessons are not free with the exception of some introduction lessons.