Like any other government-issued ID, getting a passport involves presenting several documents so you can prove your identity. In order to change your name, you need to have your original passport, a filled-out application for a new passport (see Resource section for a link to this form) two passport photographs, and an original copy of your marriage certificate, divorce record, or court-ordered name change. As of January 2010, you need a check for $120, as this is the fee for the new passport.
At the top of your passport application form, you will see an address. Write this address on a large document envelope and put the appropriate postage. Then place all your documents inside and place it in any U.S. postal service box. Make sure the envelope is sealed adequately so no one can tamper with your documents inside.
After your materials have been mailed, you can expect to wait about six weeks for your new passport to arrive. If you need it sooner, you can pay $60 (as of January 2010) for expedited service, which will get you your passport in two to three weeks. Once you have received your new passport, open it and make sure it reflects the correct spelling of your new name.