There are several things that mathematicians do:

* a *teacher of mathematics* teaches math, and you surely know some of those,
* an *applied mathematician* uses his knowledge of mathematics to help engineers, physicists, chemists, and other scientists solve math problems. For example, he helps an engineer calculate how water is going to flow around a ship that the engineer designed. Or he might help doctors calculate how a flu is going to spread and how many people will get sick.
* a *theoretical mathematician* figures out new math that nobody has done before. For example, he solves an open math problem that nobody has been able to solve. Or he thinks of new interesting math that nobody has ever done before. Sometimes he invents 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.