The game of Go has existed in something very close to its present form for nearly four thousand years but it's only been in the past fifty years that it has gained any wide spread attention outside of Asia. It is arguably the most challenging strategy game in the world to play well as its huge play area and options blow just about anything else away.
I can give you the rules for Go in a handful of sentences. The game is played on a 19 by 19 grid with players alternating placing white and black stones. The players are attempting to surround territory and any stones completely surrounded by the opponent's color are removed from the board. The game ends when both players pass and choose not to add stones (usually because there is no further advantage to be gained by playing) and the scores are essentially the area that the player has surrounded minus the number of stones their opponent has captured (there's also a handicap added to one player to allow for the advantage of going first). There's a few more rules for particular circumstances but that's the basics.
Playing Go on a computer has been a thorny problem for AI for a long time and at this point computer Go can play at the level of a typical amateur. GnuGo doesn't do a bad job at this and it's a good program to use for a player who wants to just try playing the game. Particularly helpful for the new player is the ability to scale the board down (9 by 9 is a usual size for people just beginning to play).
Most users will also want to get the glGo interface which adds a GUI to the AI provided in GnuGo. glGo also is configured for connecting to IGS-Pandanet, a popular source for playing Go online.
Go is one of the most challenging games to play that I've ever encountered. Despite the fact that the rules can be summed up in five sentences there is a great depth to the game. You could study it for years and many players do. GnuGo doesn't offer a tutorial to help you over the rough spots of starting out but it is the best Go game you're going to find for a PC.