Check specifics for the country you are in using the list at www.usembassy.gov (find it under “Citizen Services > Passports”).
In general you have to appear personally at the embassy. If possible, bring a copy of the local police report with you. (Also see, I am overseas and my passport was stolen or lost, who do I call first?) Fill in the DS-64 lost and stolen passport form as well as a DS-11 passport application form (use link below to get these forms) and prove your identity as best you can. They suggest documentation such as a birth certificate, driver’s license, photocopy of the missing passport or even an expired U.S. passport. If you have no current identification with you, you should bring along an American friend or relative with his or her passport to vouch for you. And don’t forget pictures — two (2) identical, recent color photographs taken full face, with a white background, measuring 5 cm by 5 cm (2 in x 2 in). If travel is urgent—and be prepared to prove it with your ticket—the embassy will issue an emergency 12-month limited validity passport.
For more information, visit the Travel State website