If you've lost your passport, the first step is to report your passport missing. You can go to a passport agency in person to report the passport missing and apply for a new one. If you aren't able to get there in a timely fashion or don't need a new one, you can download form DS-64, Statement Regarding a Lost or Stolen Passport, and mail it to the Department of State. Include as much information as possible and answer the questions in-depth to the best of your ability. The address is:
U.S. Department of State
Passport Services
Consular Lost/Stolen Passport Section
1111 19th St., NW, Suite 500
Washington, D.C. 20036
Once they've processed the DS-64, your lost passport will be invalidated.
When you lose a passport, the process to replace it is essentially the same as applying for a brand-new passport. You bring your proof of citizenship, identification, two passport photos and payment to a passport agency and fill out a DS-11 form. Question 18 is particularly important; fill in your name as it appeared on your lost passport, the number of the passport and the date it was issued (if you know), and write "EXPIRED" next to the Other box if it was expired.
If you're abroad when your passport is lost, you'll have a more serious issue. You must go to the appropriate U.S. embassy or consulate in the country where you are and report the passport missing. You might be able to get a temporary passport quickly, but the process varies by country. Unless you have your proof of citizenship and identification information with you, you may need to have it faxed or mailed so that you can complete the application process. Discuss any impending departures with the embassy or consulate involved and see what can be done.
To prepare for any potential problems with your passport, leave a photocopy of your passport and any identification and proof of citizenship that you don't want to take with you in a safe place with someone you trust, such as a relative or close friend. This way you can contact her in the event your passport is lost and have the information readily available to send. You could also scan your passport information onto a computer and send it to yourself via email so you can access it.
Once you've filed your DS-64, your passport will be invalid. If you subsequently find the passport, you will not be able to use it, and it cannot be reinstated. You should mail it to the Department of State. Let them know if you'd like it back, or they will simply destroy it.