According to WIKIPEDIA it is a set of partner dances, which are danced as recreation both socially and competitively.
Some ballroom dances make you feel and look like a princess and a prince dancing on the ball and those are the five Ballroom dances: Waltz, Tango, Viennese Waltz, Foxtrot and Quickstep.
(watch the 2016 World best ballroom dancing here)
Other dances make you feel like you are on a Carribean island and those dances are the five Latin-American Dances: Samba, Cha-Cha-Cha, Rumba, Pasodoble and Jive.
These 10 dances are danced all over the world in each and every country exactly same way with same steps, styling, tempo and technique, so it is rightfully called today the International Style (see the 2016 World best dancers perform every International style Ballroom dance for you here)
Two additional styles of ballroom dancing was developed and danced mostly in US: the American Ballroom, very similar to International style Ballroom) and the Rhythm, similar to International Latin-American dances.
The Smooth includes the same 4 of 5 International Style dances as Slow Waltz, Tango, Viennese Waltz and Foxtrot, but do not includes the Quickstep.
The American Rhythm includes same first 3 dances as in International style Samba, Cha-Cha and Pasodoble. The Jive becomes a bit slower and called Swing. The Rumba splited on two dances: slower version called Bolero and faster is the American Rumba. Finaly, the Mambo is a more complex competitive variation of the "Salsa on 2".
American Style dances use almost identical steps and technique as traditional International Latin-American and Ballroom dances but have less restrictions in both hand hold and positions.
The very beginner do not have to care about the difference between two styles. Whatever style teacher knows better.