Fill out form DS-5504. You may obtain it from your local passport agency or acceptance facility, print it out online and complete it by hand, or even complete it online and then print it. Don't forget to sign the application on page 1.
Submit the completed form and supporting documents by mail to the address indicated on the form. You will need to submit the completed form, your current passport, a new passport photo, and evidence to support the requested change. For example, if you want to replace your maiden name with your married name, you should submit a copy of your marriage certificate.
If you are in a hurry, write "expedite" on the outside of the envelope, and enclose a $60 check made payable to the U.S. Department of State. You may also send the envelope via overnight delivery service to the Passport Agency, so that it might be processed faster.