There are 5 Ballroom dances that are a part of the international curriculum and which are used commonly for both competitions and social dancing. There area a variety of different schools and methods for all of these dances, but it is the international style I will be focussing on. They are:
The Waltz is done to music that is in ¾ time. This means that instead of counting to 4 (or 8) before starting again, the patterns in the music count to 3 (or 6).
The Foxtrot comes in two main varieties, the Slow Foxtrot and Social Foxtrot. Social Foxtrot is simpler and easy to apply to many different pieces of music. The Slow Foxtrot has more complicated footwork and styling.
The Quickstep is a high energy light on your feet dance whose basics have similarities to the Social Foxtrot (at a higher speed!).
Though it seems like it should be a part of the Latin Dances, Tango is, in form more like the Ballroom dances. Like the others mentioned, it has a fixed hold and moves around the floor.
Lastly, the Viennese Waltz is a stylized version of the Waltz mentioned above. It is usually done to 6/8 music which means that the tempo is much faster than the traditional waltzes. As the footwork is faster, many of the things encouraged in traditional waltz, such as rise and fall, are discouraged.