Ultimate is certainly one of the best known frisbee games, at least here in the US. It's sort of a cross between basketball and American style football. The goal is to to catch the frisbee in your opponent's endzone, somewhat like scoring a touchdown in American football. When you have the frisbee, you can pivot on a foot but can't take any steps, sort of like basketball.

Ultimate is a non-contact sport; it's considered a foul to touch the player holding the frisbee.

The official rules are on the USA Ultimate web site.