Different people attend graduate school at different points in their lives. Some students apply in the fall of their senior years for admission to graduate school in the next fall term. Other students work in business, industry, or government for a few years, and then apply to graduate school on a part-time basis, often with financial support from their employers. Others choose to return to school full-time, after a few years of work.
Letters of recommendation and scores from the Graduate Record Examination (GRE) are required for admission to graduate programs. Several of the letters should be from faculty members who know the student's work in advanced mathematics courses. You would normally take the GRE General Test and the Advanced Mathematics Test during the fall before you plan to enter a graduate program, usually no later than October and November. Information about the GRE test is available on-line.
Deciding whether or not to attend graduate school is a major decision and you should take care not to discount the possibility of going on to graduate school too early in your mathematical studies. The more abstract emphasis in certain courses-Math 211, 214, 307, and 311, for example-might be somewhat disconcerting upon first acquaintance and students sometimes rule out graduate study on the basis of their first contact with abstract mathematics. Such a snap judgment is usually a mistake.
At least one of Math 307 and Math 311 should be completed no later than the your junior year. This allows you to make a more informed judgment about applying to graduate school in the fall of your senior year. It is certainly not the case that graduate mathematical studies all deal with theoretical mathematics. However, even in the most applied parts of the mathematical sciences, students will encounter a progressively more abstract emphasis as they delve deeper, and you need experience with several theoretical courses before making a decision about further study. In addition, graduate admissions officers, particularly in doctoral programs, want to see the results of courses like Math 307 and 311 before making admissions decisions.
You should ask faculty advisors for advice on choosing a graduate school. Depending on your mathematical interests, some programs will be more appropriate than others, and faculty advisers can help you judge the relative strengths of various programs. One place to start comparing graduate programs is the annual AMS publication "Fellowships and Assistantships in Mathematics." It lists mathematical sciences graduate programs and shows the areas in which each specializes by giving the broad research areas studied by recent doctoral graduates. Just as important, it gives details about financial support available for graduate study at various universities. Another important source of information is the periodic National Research Council report on the quality of research graduate programs. The American Mathematical Society publishes rankings of mathematics doctoral programs based upon these reports. The magazine US News & World Report also ranks PhD programs in mathematics at its web site.
It is an unusual senior in college who knows for sure what he or she will choose as a dissertation area after two or three years of graduate study. It is usually unwise to choose a graduate program because it specializes in a part of mathematics that interests you today. As an undergraduate, your exposure to different aspects of mathematics is still quite limited. One reasonable approach to address this issue is to choose a large, high quality program (as judged by the National Research Council). Such a program will offer a wide spectrum of good choices for an eventual research area.