Make sure you're eligible. You must be a citizen of a country other than the United States at the time of application; have a Ph.D. or equivalent professional qualifications; have proficiency in English; and be in good health.
Contact your country's binational Fulbright commission to obtain an application. Or, if your country doesn't have a binational commission, contact the U.S. embassy in your country.
Decide whether you're more interested in lecturing, in pursuing your own research, or both.
Think carefully about what sort of program you'd like to design for yourself and about how you'd describe the project. Design your program around a theme of mutual benefit to the United States, your home country, the U.S. institution and your home institution.
Take steps to affiliate yourself with a U.S. institution, either by getting a letter of invitation from the institution or by working through your country's binational Fulbright commission or U.S. embassy.
Prepare the following materials: the application form; a project statement; a curriculum vitae (including a list of publications); three letters of reference; a letter of invitation from the U.S. host institution (if available); and a certificate of health.
Notify professors, colleagues, former employers, etc., well ahead of time for any letters you need from them.
Send in all application materials before the deadline, but remember to photocopy your materials first and keep a copy for your records. Deadlines vary on a country-by-country basis.