The American Mathematical Society maintains an extensive web site providing career and job information, including academic and non-academic jobs and a page aimed at graduate students and recent Ph.D.s The Society for Industrial and Applied Mathematicians also has a Careers and Jobs page. Extensive discussion of the applications process and other issues can be found at Concerns of Young Mathematicians.
The AMS maintains an automated database of job applicants with advanced mathematics degrees MathJobs.Org for submitting job applications.
Applications for academic positions for the following academic year are usually due between November through February. Many job listings are posted at the AMS employment center. The application often involves a cv, a research statement, and sometimes a teaching statement. Samples of these can be obtained from the faculty or more senior graduate students.
Research positions often ask for three letters evaluating your thesis research, and one letter evaluating your teaching effectiveness. It's a good idea to think about who will write your letters well in advance. Often, these will be the same people on your thesis committee. However, it often helps to have an international aspect to your research letters. If you go to an international conference before going on the job market, you should talk with the other mathematicians in your field by asking questions etc. If someone is interested in your research, they are often glad to write a letter. Requests for recommendation letters should be made at the beginning of the semester, preferably at least two months before the letter is due. Often more senior mathematicians have forty or so letters to write each semester, and asking at the last minute can mean that you may end up with less consideration.
What happens after you file the application varies drastically on the kind of position. For some research positions, no further interaction may take place before a decision. For more teaching-oriented positions, you may be asked to give a colloquium, a research seminar, and/or teach an undergraduate class. If you are invited to give such a talk, you should arrange to give a practice talk, and collect constructive criticism. Please get in touch with Professional Development Coordinator to arrange an audience. After an invitation, it is a good idea to enquire about the background of your audience. Is it research mathematicians? Undergraduates with no mathematics background? If you are invited to give a colloquium talk but not teach an undergraduate class, then often your effectiveness at communicating your subject will be used to estimate your teaching effectiveness. This makes it especially important to spend the majority of the time aiming at an audience with no background in your field.
Other sources of support
The NSF offers Postdoctoral Fellowships in the Mathematical Sciences with deadline in October for support the following year. The Clay Institute has a Liftoff program for summer support following the Ph.D., with application due in February. candidates should be nominated by the department. The Sloan Foundation and American Institute of Mathematics also offer support for mathematicians who are about to finish or have just finished their dissertations.