All institutions of higher learning require math professors to perform a variety of roles. Teaching, of course, is almost always in the job description of a math professor. But research and service also are required. The emphasis varies according to whether the school sees its mission as one of primarily teaching, primarily research or both. Professors at top schools often must do leading research that is particularly novel and well-regarded as well as teach and serve in other more ostensibly mundane roles. The lower you go on the food chain in academia, the less important research is.
The most competitive schools and the hardest to be a tenure-track math faculty member are those where highly accomplished and noted authorities have taught and established strong programs in their disciplines. At these schools, continued accomplishment in original research--solving previously unsolved problems--that can be described as novel and far-reaching is a basic requirement for getting tenure. Such institutions also impose demanding teaching loads, though lower courses may often be picked up by junior members or even teaching assistants.
Less prestigious institutions also require mathematicians to distinguish themselves in research. In fact, the difference between these schools and their most elite counterparts as far as research goes can sometimes be hard to detect, particularly for a junior faculty member trying hard to get a permanent position. At four-year institutions that have a history of emphasis on teaching, however, it is understood that the research component of the job is less important. But an enterprising young mathematician can still distinguish himself or herself by achievement this way.
It is the rare math professor who doesn't have to teach. At a typical university with a graduate program, most professors pick up three to five courses at a time, perhaps two undergraduate courses and one graduate course. Heavier teaching loads are usually expected at institutions where research is of lower importance. At community colleges, math professors may teach five entry-level courses and one calculus course and do no research at all.
As a teacher, a professor must plan lessons, lecture in class, assign and correct homework, write and grade quizzes and exams and hold office hours. Competence and confidence in a wide array of mathematical disciplines from statistics to calculus to higher algebra are expected. It has also become common for students to expect professors to be available by email or online at odd hours, although this is not required.
All faculty members are required to provide service in the bureaucracy of the department. This means serving on committees or being involved in the development of curricula. A typical example might be the undergraduate calculus sequence committee in which everything from the text to whether students must learn how to prove any theorems is decided. A less formal kind of service is when professors become mentors to students.