Updated 6/6/2010: Aso Brain is currently not accepting new registrations and many older accounts have been purged. It isn't clear when (or if) the ban on new registrations will be lifted.
Aso Brain Games has some very cool, family-friendly Java based games on their site. My favorite is a version of Settlers of Catan called Xplorers. If you haven't played, it's a card-based resource collecting, city building game. Players collect resource cards that they use to build roads, towns, and other infrastructure. If your family has more than one computer online, you can play against one another. You can also play against other online players or against the computer. Although the content is family-friendly, engaging in multi-player games online does open you up to possibly inappropriate comments. The game has the option for privacy, so no one can join or comment.
There are three other games on the site, including Match! which is like the card game Set.