Most passport applications are processed over a few weeks, which means the applicant will not receive the actual passport for four to six weeks. People who need an application faster than that need to request one through the channels established by the Department of State.
Visit travel.state.gov and get all of the materials necessary for a passport. This will include an application form, which in most cases is DS-11. See the link to "First Time Applicants" for more information.
Take all of the necessary materials to a passport agency. You must sign form DS-11 at that office.
Request an expedited service and pay the fee. As of 2011, the fee was $60. It will take two weeks to get a passport this way. If you must leave the country within a few days, call the National Passport Information Center and request an appointment with an agent.