There are several things that mathematicians do:

*teachers of mathematics* teach math, and you surely know some of those,
*applied mathematicians* use their knowledge of mathematics to help engineers, physicists, chemists, and other scientists solve math problems. For example, they help engineers calculate how water is going to flow around a ship that the engineer designed. Or they might help doctors calculate how a flu is going to spread and how many people will get sick.
*theoretical mathematicians* figure out new math that nobody has done before. For example, they solve open math problems that nobody has been able to solve. Or they think of new interesting math that nobody has ever done before. Sometimes they invent new math that allows physicists and other scientists solve physics problems that they could not solve with old math.

Most mathematicians do a bit of everything. For instance, I teach at a university, I do theoretical math, but I also do applied math because my research is about math and computers. So I often figure out how to program computers so that they can do math by themselves, which means that I need to write programs as well as invent new math that allows me to write even smarter programs. If you would like to know examples of this, just ask.