Visit or call the American Embassy in the country where you are traveling as soon as possible. According to the State Department website, you will need to work through the American Citizens Services unit of the Consular Section of the embassy to prove your citizenship status and obtain a replacement passport.
Take any identifying information you have with you when you visit with a consular agent. This should include any identification cards, including a driver's license. You may also need to provide a witness to verify your identity. This can be someone in your traveling group, or a member of your family on the phone from home.
You will not be permitted to fly out of the country or reenter your home country without a passport. There will be a cost associated with the new passport and it may take a couple of days to issue, depending on the situation. The sooner you can report the problem, the better.
