This page describes typical routes to a career in research mathematics, concentrating on pure mathematics and jobs in academia.
Becoming a math professor in one simple paragraph The first step is lots of school: 4 years as an undergraduate, typically followed by 5 years of graduate school, resulting in a PhD. Then often comes 2 or 3 years of postdoctoral training (the "postdoc"). After the postdoc (or immediately after graduate school if not doing a postdoc), mathematicians who want a career in academia look for tenure track positions at a college or university. Academic jobs involve a mix of teaching, research, and service; the precise mix varies widely among different colleges and universities. After 3-7 years (usually less if you did a postdoc, more if you didn't) you come up for tenure. If you are successful then you have a "permanent" job as a professor.
Many mathematicians work for the government or various sectors of business and industry. The educational training is similar to the description above, except that the choice of classes and specializations may be different, and it is less common to do a postdoc. And the concept of "tenure" does not exist outside of the teaching professions.
Undergraduate studies Most math majors take one math class each semester for the first year or so, and then two classes for their remaining undergraduate years. The first few classes are usually calculus, differential equations, and linear algebra. After those classes there are some options depending on whether the concentration is going to be in pure or applied mathematics.
In pure mathematics the focus quickly moves to writing proofs. The style of research mathematics is Theorem-Proof-Theorem-Proof, where "Theorem" refers to the precise statement of a mathematical result, and "Proof" refers to the logical argument which establishes the truth of the theorem based on previously established results.
In contrast to the proofs of high-school geometry, the proofs of abstract mathematics have the elegance of a poem -- the good proofs do, anyway.
Basic classes in the pure mathematics stream include abstract algebra, real analysis, topology, and complex analysis.
Basic classes in applied mathematics include partial differential equations, combinatorics, complex analysis (the same topic as in the pure stream, but sometimes taught from a different perspective), and mathematical modeling.
It is becoming more common to do "undergraduate research," often as part of a Research Experience for Undergraduates (REU) program sponsored by the NSF.
Graduate studies Graduate school in mathematics usually begins with more emphasis on course work, and ends with more emphasis on research as the doctoral candidates work on his or her thesis. The "thesis" is the document which contains the original research of the candidate. In order to receive a PhD in pure mathematics, you have to prove new mathematical results; those results are presented in the thesis. The last step to a PhD is submitting the thesis to an examining committee and giving a public talk describing the results.
A masters degree is usually skipped or viewed as a formality for graduate students pursuing a PhD in mathematics.
In the first year or two, most mathematics graduate programs require comprehensive exams (also called preliminary exams or "prelims") In pure mathematics the (usually written, but sometimes oral) exams cover the core areas of Real Analysis, Complex Analysis, Algebra, and Topology. In applied mathematics programs, there is a greater variety of topics covered in the exams. Many programs have a second round of exams a year later, covering specialized material which is designed to determine if the student is prepared to begin work on a thesis problem. The second round of exams is almost always an oral exam, administered by a committee who keeps asking probing questions until you reach the end of your knowledge.
After (usually before, unofficially) passing all the exams, the doctoral candidate finds a thesis advisor. This requires a complicated balance that, when successful, produces a good match in terms of both mathematical interest and personality. This is a significant decision because the thesis advisor will suggest (or help the student find) a suitable problem whose solution will become the student's thesis. The advisor has a dominant influence on the research direction of the student, will write many letters of recommendation over the coming years, and pretty much is the determining factor on whether the students is successful. After graduation, the student will forever be known as "a student of [thesis advisor]."
After the exams, the student spends the next year or two (or three) working on a thesis. The term "dissertation" is used in many subjects, but in math the more common term is "thesis." It is common to also take a specialized course each semester, and also to participate in a weekly seminar. But working on the thesis is the dominant activity. Note that the verb is working, not writing. The actual writing of the thesis does not begin until after it is clear that the hoped-for results have actually been established. It is not uncommon to find a mistake and have to start over again.