Here are some resources to help you find an academic job in pure math.
Announcements of open jobs can be found through the AMS employment services website. In particular there are announcements in the EIMS database, and in mathjobs.org. You will need to create an account at mathjobs.org. There are also announcements on many departmental webpages. I'd recommend that you check the web page of your dream job separately, just to be sure. The Chronicle of Higher Education, the MAA (Mathematics Association of America), the AWM (Association for Women in Mathematics), and the CMS (Canadian Mathematical Society) all have job listings. There is overlap between these listings, but you should check them all. If you're interested in jobs abroad, the timetable is often different than the North American one. You may also be interested in applying to visit one of many research institutes. Many of these do not advertise openings, but have application information available on their webpages. Most deadlines fall between the first of November and the first of January.
The AMS joint meetings (JMM) are held in early January every year. The Employment Center at the JMM is a part of the JMM arranged to match potential employers with job seekers. Many initial interviews are set up through this service. The deadline to sign up is very early in the Autumn. Here's an overview of the services. I'd also recommend trying to give a talk at the meetings. You can either apply to talk in a releveant special session, or apply to give a general contributed talk. Potential employers often attend these talks.
Most jobs can be applied for through mathjobs.org, but some institutions have their own web based application process, and you may need to apply for some jobs through standard post.
The NSF has postdoctoral fellowships. You need to choose a sponsor who you'd like to work with. As part of the application process, your sponsor will have to write you a letter. The deadline is in mid October.
You will need to prepare a CV, an AMS coversheet, a research statement, a teaching statement, and cover letters. In addition, you will have to obtain several letters of reference about your teaching and research. (The number and distribution depend on what type of job you are looking for.) You may also need to prepare some additional material, such as a publication list. I recommend setting up a web page if you don't already have one, and posting your application information and supplemental information (such as preprints) on the page.
Below are a some links to advice for job seekers that you may find useful.
The AMS's compilation of advice for new PhD's.
Terry Tao's career advice.
AWM's career resources page.