American style is danced primarily in, America.
In the US, American style tends to be used more often for social dancing while International style tends to be used primarily in competitions. This doesn't mean that you can't compete in American style or dance International style socially. As for the dances themselves, they are grouped into two categories for each style. In American style, the categories are called Smooth and Rhythm and in International style they are called Standard and Latin. For the most part, the Standard and Smooth categories contain the same dances and the Latin and Rhythm categories contain basically the same dances.