Mathematics is the study of how our universe operates. Math is a way to interpret nature using ideas like calculus, rings, topology, or lie Algebras. Math is based on logic; any mathematical proof is a series of logical steps that arrive at a certain conclusion. But, this "definition" of math is probably lacking in some way. Math encompasses so much, it's hard to describe all of what it is. So how about I describe parts of it, from my experience.
For most of the math classes I've taken, I was given a set of elements, a couple of operations, and some basic definitions (for instance a ring). Then the rest of the class was just exploring that set of elements by understanding different properties (if the ring was a field or commutative) and theorems (the isomorphism theorems of rings). That was honestly modern algebra I and II, linear algebra I and II, and topology. But, then you have your calculus and advanced calculus classes. There, you would study limits, sequences, series, continuity, derivatives, integrals, etc... Then of course you have probability and statistics. I could go on and on. But, to better understand what math is, study it! Learn about the different areas of mathematical research (algebra, geometry, differential algebra, algebraic topology, etc...). Learn about the countless properties of the real and complex numbers. The study of math is so vast. One thing that got me more interested in mathematics was its history.
Here are some of my favorite moments/discoveries in the history of mathematics. One of the most important mathematical works is Euclid's Elements. Euclid essentially took all of Greek geometry at his time and made it into a book. He had his axioms/postulates from where he started and then came up with many geometrical theorems and properties. Then of course we have the discovery of calculus by Newton AND Liebniz in the 1700s (or 1600s I think). After the discovery of calculus, the study of physics progressed rapidly. And then we have the mathematical discovery machine in the great Leonhard Euler. He was born a genius and had John Bernoulli as a mentor growing up. Euler made contributions in all areas of mathematics: complex analysis, calculus, geometry, number theory. Even after his death he had some of his works still being published. Euler is my favorite mathematician. I also would like to mention Rene Descartes. From what I've read, much of our algebraic notation we use today is from him. One last mathematician I would like to mention is Evariste Galios. I will not spoil everything for those who have not heard of him. Look him up, his story is astounding.