If you have changed your name, you may apply in person for a new passport to be issued in your new name. Airlines often require that the name in the passport match the name on the airline tickets.
You will need to submit documentary evidence such as a marriage certificate, divorce decree, court order, royal decree, or other satisfactory evidence to support a change of name. Only originals or certified copies will be accepted. Documents presented must be in English.
In addition to the above, follow the instructions for obtaining a passport for an adult, or minor age 16 and over, or a passport for a minor under age 16. Note that if your most recent passport was issued within the past year, the Passport Wizard in the above links will help you complete form DS-5504 and there will be no charge for the passport.