One of the things that struck me about DanceSport was how utterly arbitrary some of it seemed. Why those dances, and how did they come to be defined the way that they were?
As you might imagine, there are bureaucrats and committees involved. Without going too much into that, the result is a taxonomy of sorts for ballroom dance.
Of course we Americans can't agree with the Europeans on what is what. They have "international" style, vs. our "American" style. Although some names are similar (foxtrot, tango, rumba), the dances are significantly different. The international style typically is less permissive, and if Dancesport achieves full Olympic status, it will be their variety.
Also, there are "smooth" or "standard" dances and there are "rhythm" dances. Examples of the first two are slow foxtrot, waltz, quickstep, Viennese waltz, and tango. For rhythm dances, examples are rumba, bolero, mambo, merengue, samba, cha-cha (or cha-cha-cha in Europe), paso doble, east coast swing, west coast swing, hustle and jive. The smooth dances are snootier and competitions are done in evening dress. Rhythm dances IMO are lot more fun to watch and to do, and the women dress very provocatively.
Some of the dances are "spot" dances - the dancers remain in a fairly limited area. Others, typically the smooth dances but also including samba, are expected to travel around the floor counterclockwise.
Rumba, slow foxtrot, bolero, and waltz are done at slower tempos. East coast swing, cha-cha, quickstep and Viennese waltz are done at fairly frantic tempos, and the latter two at times are almost like a three-legged sprint. Others fall in between.
They have different attitudes too. Foxtrot is "cool". Bolero, waltz, Viennese waltz and rumba are romantic. Tango is...tango. Paso doble is bullfighting - the man is the bullfighter and the woman is his cape. Cha-cha is flirty and mambo leans toward raunchy (think "Dirty Dancing" to the theme from "Sex in the City"). East coast swing and jive are just, well, hell-raising.
All of them have their own type of music. Waltzes and Viennese waltzes have time signatures that are multiples of 3, and the rest are normally in 4.
Suppose you want to give this a try. The easiest ones to start with are probably foxtrot and rumba - neither is particularly fast. East coast and west coast swing are the most popular, but are more energetic and so might not be the place to start if you're in really bad shape. I'm speaking of the American style - I haven't tried the international stuff.
If you want to try tango, be aware that there are 3 flavors - American style, international style, and Argentine. The last is the most authentic but also the hardest to find instructors, partners and competitions.
That ought to keep you busy for a while. Give it a try!