The U.S. Department of State issues passports to U.S. citizens, so that is the agency you will apply to for a passport. Whether you need your passport instantly or in two months, the application form and the documentation you'll need to provide are the same.
You will have to submit your completed Form DS-11: Application for a U.S. Passport, and provide your Social Security number and documentation proving your citizenship. The latter can be an official copy of your birth certificate, an old, undamaged passport, an official certificate authenticating your birth to U.S. citizens abroad, a naturalization certificate or a certificate of citizenship.
You will also have to provide identification, which can be a current valid driver's license, military or government identification card or a naturalization certificate. You will have to permit the official to make a copy of this identification to submit with your application.
You will have to include in your application two passport photos, which are two identical color photos, 2 by 2 inches in size, of yourself dressed in ordinary street clothes, taken within six months of the date you apply.
At this point, the application process for getting a passport fast differs from a routine application. If you already hold a U.S. passport and you had ample time, you could file an application for a renewal by mail, but if you've never applied for a passport before, and/or you need it fast, you will have to apply for it in person.
If you plan to travel internationally within two weeks of your application, or if you need your passport four weeks prior to traveling because you have to apply for a visa, you should by-pass your local passport application center and go straight to the nearest regional passport agency. You can find a list of regional agencies on the U.S. State Department website, and you can also schedule an appointment at one online.
You will pay significantly higher fees to expedite the processing of your application. Ordinarily, if you're over 18 and you renew a passport by mail, the application fee is $75; if you apply in person, the fee is $75 plus an execution fee for the local facility of $25.
For expedited processing, you'll pay an additional $60 plus any overnight delivery charges. The U.S. State Department recommends strongly that you opt for overnight service of your documentation to the processing center, and for return overnight delivery of the completed passport to you.
While routine processing time for a passport presently is taking about four to six weeks, expedited service takes two to three weeks. If you haven't chosen to have the passport sent back to you via overnight courier, you will need to factor in additional time for U.S. mail service.