Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
-- Aristotle

Mathematics is a language invented by humans to describe certain objects and processes. Evolving from counting, simple arithmetic, and geometry, it has blossomed most wonderfully into set theory, boolean logic, calculus, and many odder disciplines. From the beginning, math has been fueled by smart, obsessive, and bored people who had enough leisure time to discover that universe was filled with odd regularities.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
-- Aristotle

Math has the distinction of being the only a priori science (unless you consider philosophy to be a science). Visual aids and experimentation can certainly help, but most math is done on paper. None the less, it has great worldly benefit, from architecture to zoology. It's fun to play with -- magic squares, sudoku, the Prisoner's Dilemma, the Monty Hall Problem. It's important to our lives -- it programs our computers, launches our spaceships, and encrpts our emails.

The whole is more than the sum of its parts.
-- Aristotle

In theory, math can be applied to describe any observable phenomenon that exists, and many that don't. But Godel's theorem shows that it can't actually describe every observable phenomenon, because any given system can't describe itself. This has not yet caused math to fall out of fashion.

To Thales the primary question was not what do we know, but how do we know it.
-- Aristotle

There is, to my knowledge, no formal categorization of the subfields of mathematics, and no clear hierarchical structure between the 'main fields' of mathematical study and the 'sub-fields'. But here is a list of some of the main branches of mathematics.

Needless to say, you should learn more math. You should node more math. Unfortunately, while E2 is an excellent place to node math, it doesn't actually do a very good job of teaching it, unless you've gotten a good foundation elsewhere. Fortunately, your local library will have the primers to get you started.

Mathematics is a "formal" discipline formed around a human cognitive mechanism which accounts for perception of discrete entities (and by extension, quantitative relationships). It should be noted that discrete entities and quantitative relationships do not actually exist in nature per se; rather, we accept the notion that they do as part of the natural reductive process of our consciousness (a sometimes-convenient compromise). In this context, Mathematics is a (characteristically exhaustive and rigorous) compendium of metaphors based exclusively on discrete and quantifiable subject matter (i.e. numbers), where that description is itself, as much as possible, discrete and quantifiable.

Kurt Godel's Incompleteness Theorems (published 1931) correctly identify that any formal system (i.e. Mathematics) is inherently flawed, specifically "in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system." To restate, Mathematics is effectively a study of the consequences of the particular compromises inherent in human cognition. Caveat emptor.

I believe that mathematics can best be defined as the symbolic manipulation of ideas. This definition could be interpreted to include such disciplines as literature, art, and philosophy. At its core, mathematics has more in common with literature, art, and philosophy than it does with the sciences, with which it is more often linked. This can best be explained by the fact that the qualities that make one poem better than another are the same as the qualities that make one proof better than another. The ultimate goal of all poems and proofs is to say more with less.