Passport policies and processes change, so we strongly suggest checking the latest on the Bureau of Consular Affairs' U.S. Passports and International Travel website — the source of all the info below.
For passport first-timers, your tour should start with a visit to your nearest passport agency or acceptance facility. The same holds true for minors age 15 and under (or if you received your previous passport while that age); if your last passport was lost, stolen or damaged; or if you ordered your old passport more than 15 years ago. A name change that you cannot legally document will also require some quality time at one of these services.
Of course, you'll want to gather a few necessities before you go: ID (along with a front-and-back photocopy that you can leave with the agency); proof of citizenship (originals, not copies); a passport photo (color headshot, 2 x 2 inches [51 x 51 mm]); and some way to pay the fees (options vary). You can fill out the passport application, Form DS-11, in person, or download and print it ahead of time, but you must not sign it until told to do so by an acceptance agent.
ID options comprise a valid driver's license, government ID or military ID. Proof of citizenship includes a certified birth certificate or Consular Report of Birth Abroad. A valid passport, or a naturalization or citizenship certificate, counts for both.
If you don't have any of the IDs listed, gather as much proof as possible and find a citizen who has known you for at least two years and will swear to your identity. Applicants who cannot provide current primary evidence of citizenship can, for a fee of $150, ask the bureau to perform a search for their previous U.S. passport or Consular Report of Birth Abroad (if applicable).
Some additional or special rules apply to minors 17 and under; to diplomatic, official and regular no-fee passports; to people applying from outside the U.S.; and to people going through gender transition. Legal problems, such as failure to pay child support, can also impede your ability to obtain a passport.
If these don't apply to you, cheers! You can most likely renew by mail. As an added bonus, you can apply for a passport card by mail, too, even if you have not had one before. Simply fill out your renewal form, Form DS-82: Application for a U.S. Passport by Mail, and include it along with your old U.S. passport book and/or card, your fee and your passport photo. If you've changed your name, include a certified copy of your marriage certificate or court order.