gpokr.com is a monthly Texas No Limit Hold’em Poker competition. The object is to win as many chips as you can over a 1 month period. You start with $1,500 in chips and if you bust out you can refill. There is a very friendly community there and it serves as a good intro to poker.
kdice.com is a game of dice based risk, you compete with 6 other players at a time to dominate a map by linking your territories. A good performance gains points and again culminates in a monthly comp with the aim to accumulate as many points as you can.
Both these games run in your browser without any downloads and free and lots of fun.