From my, admittedly limited, interaction with mathematicians in my life and a bit of extrapolation:
- Academia: teach advanced mathematics and do research in mathematics for a university. There's still lots of unsolved problems in math and also plenty of overlap with computer science, which also has lots of research possibilities
- Public sector: governments of all levels need at least statisticians, if not more specific mathematics skills depending on what they're trying to do (e.g. research, engineering, economics, etc.)
- Private sector: lots of engineering companies employ a few mathematicians or at least physicists who are really good at math to make sure their next bridge/plane/ocean-boiler will actually work
There's a lot of overlap between all three but I roughly split them up based on where I'd expect the majority of jobs like that would be (e.g. I'm sure NASA employs a good deal of mathematicians, but so does Lockheed Martin and friends). Also a lot of people get a degree in mathematics and then specialize further with a masters and/or doctorate in computer science or physics, since both of those can be quite math-heavy and are better-funded fields.