No matter how you categorize it, it is all ballroom dancing, known competitively as Dancesport. There are two styles to dance in.
American Style, which is smoother, less rigid and open, meaning that you can stray from the syllabus (we'll get to that later); and International Style, which is the only one danced competitively outside the US, is more difficult because it requires more agility, straying from the syllabus is not really allowed (until you get to an "open" level of performing) and looks a lot harder to do but really awesome nevertheless. Each of these two styles is broken down into two categories: traveling dances and spot dances.
The traveling dances in American Style Dancing are called "Smooth" and are: Waltz, Foxtrot, Quickstep, Viennese Waltz and Tango; the spot dances in American Style Dancing are called "Rhythm" and are: Rumba, Cha cha, Mambo, Bolero and Swing. The traveling Dances in International Style Dancing are called "Standard" and are: Waltz, Foxtrot, Quickstep, Viennese Waltz and Tango; and the spot dances in International Style Dancing are called "Latin" and are: Rumba, Cha cha, Samba, Paso Doble and Jive. The dances in American Style Dancing are supposed to be the same as the International Style dances because they are modeled after them, save Samba/Mambo and Paso Doble/Bolero. Those are just two switches they made for some reason. Also, the terminology is basically "Standard" for any of the traveling dances, regardless of which style and "Latin" for any of the spot dances, regardless of which style.
The Standard dances are danced with straighter hips, more rigid movements and (almost always) a closed frame (meaning that the two dancers--leader and follower--don't break the hold). The Latin dances are danced with more hip action referred to as "Cuban motion" (referring to the making of a sideways figure 8 with the hips) and, in the higher levels of dancing, are danced quite often in open frame (meaning holding at most, one side of the frame up, typically just holding hands in some form). Each of these dances has a syllabus, a list of moves that are learned and subsequently put together in what are called "amalgamations" meaning combinations of the moves. Each of these syllabi is then divided into level of competition.
The order is as follows: newcomer/beginner, bronze, silver, gold, novice, pre-champ, champion, professional.
So, short answer:
There are two types of dances when it comes to ballroom dancing: traveling dances and spot dances. The traveling dances are those labeled Smooth and Standard but most often referred to as Standard; the spot dances are labeled Rhythm and Latin, most often referred to as Latin and are danced competitively in two different styles.