What are the different subsections in applied and theoretical mathematics?

I'm answering this because there are no other answers so far and because one of my majors 35 years ago (ugh!) was mathematics. But I'm sure a current, practicing mathematician could give you a much more complete answer.

At that time, the big courses included abstract algebra (group theory and its relatives), classical/real analysis, and topology, with smaller siblings complex analysis, number theory, set theory, combinatorics, probability, statistics, and so forth. The latter pair are closer to applied. There were professors specializing in various of those areas, but I couldn't give you the names of the individual specializations themselves.

More recently, one of the folks I follow (John Baez, http://math.ucr.edu/home/baez/ , https://plus.google.com/u/0/+johncbaez999) has been working on and blogging about something called category theory (which frequently sounds very theoretical but has real applications, and not just to physics). A grad-student friend of mine, Curtis Bennett (now at CSU Long Beach), worked in an abstract area of geometry and topology called "buildings" (https://en.wikipedia.org/wiki/Building_(mathematics)), but that's the limit of my knowledge there.

As I said, a real, practicing mathematician could give you a much better answer, but perhaps this will serve as a starting point. (Note, however, that you'll have advisors in college who can help directly; you'll also have access to all of the professors, postdocs, and grad students, most of whom will be doing active research in various areas and some of whom may employ undergrads as research assistants.)