The notion of "Ballroom Dance" depends on the epoque. We all know of or heard of balls that featured Minuet, Quadrille, Polonaise, Pas de Gras, Mazurka and many other exciting dances, which alas! today are placed into the category of Historical dances.
In times past ballroom dancing was "social dancing" of privileged classes, leaving "folk dancing" for other folks. Today ballroom dancing is much more democratic, and the boundaries between once polarized ballroom and folk dances become blurred. Notice however that even in times long gone many "ballroom" dances were 'elevated' folk dances.
Today one may speak of "Strictly ballroom" dancing, with its competitions, schools, societies, books of technique, and of "Social Ballroom" dancing, with emphasis on having fun.
Contemporary ballroom dance technique has been extensively studied and formalized. Medal Exams are a commonly accepted standard of measurement of a dancer's technique according to conventional standards, franchise studios in the United States classify them as Bronze, Silver, Gold and Gold Bar for the social dancers. For amateur competitive dancers the rankings go Bronze->Silver->Gold->Novice->Prechampionship->Championship (roughly equivalent to the A->E rankings in Europe and Australia), then Rising Star and Open Professional for the pro ranks. The International Olympic Committee recognizes competitive ballroom dance as a DanceSport.
Most ballroom dances were social and/or folk dances before being formalized as ballroom dances, and many of these dances are still danced as social and folk dance.
Coming from grouping dances in competitions, the following divisions of contemporary ballroom dance are recognized: International Standard and International Latin. In addition, American Smooth, and American Rhythm are widely popular in the USA. The former two divisions are called International Style and the latter two are American Style.
As you may see below, both styles include dances with the same names, however they are danced quite differently. Therefore in discussing dance technique, the dance is named including its style, e.g., it is spoken of American Style Rumba vs. International Rumba or American Tango vs. International Tango. In a way, "Standard" matches "Smooth" and "Latin" matches "Rhythm".
Standard and Smooth dances are characterized by travelling: couples travel around the dance floor (along the line of dance, typically counter-clockwise). Time may be 2/4, 4/4 or 3/4.
Most "Latin" and "Rhythm" dances are Stationary, i.e., do not travel much, with the exception of Samba and Paso Doble. Time is 2/4 or 4/4.
Of course, all the above can be and is danced socially in numerous dance clubs, schools and studios.
In addition, "Social ballroom dancing" recognizes the Nightclub Dances category. Nightclub dances are less formalized than the other ones. A number of them are proudly called Street dances. Nightclub dances are danced differently in different places and club/street styles differ from styles taught in ballroom studios.
Pretty close to "Ballroom dances" and right beside "Nightclub dances" stand Country/western dances, danced both competitively and socially at C/W bars, clubs, ballrooms.
Now that we have ventured so far away from "true ballroom", I am tempted to consider one more category: Regional Ballroom Dances. I will start filling it with the subcategory of Cajun Dances blossomed in New Orleans, with sprouts reaching both coasts of the USA.