Weiqi aka Go aka Baduk.
It's not a computer game, but computers can play it (they are currently hideously strong and getting stronger). You can play face to face, offline or online.
http://www.britgo.org/
You can play on boards of many sizes, 19 * 19 is standard, 13 * 13 is good for beginners, 9 * 9 is very tactical, very strong players sometimes play that against each other, I think the emphasis on tactics of the 9 * 9 game is too much for beginners, but your mileage may vary.