You can renew a passport that's expired, as long as it meets certain requirements. First, it must be undamaged; you will submit it with your application for a new one. It must also be less than 15 years old -- which it is, if it expired recently -- and it must have been issued when you were 16 or older. If your name has changed since it was issued, you'll have to furnish legal documentation for your name change with the renewal application.
If your passport meets the renewal requirements, you can apply by mail. You'll have to include your expired passport along with the required documentation. This includes your new passport photo, fees and a completed DS-82 form, available both online and at passport acceptance facilities. If your name has changed since you got your last passport, you must also include a copy of your marriage certificate or court order documenting the change.
If you do not meet the requirements for mail-in renewal, or if you lost or damaged your expired passport, you must apply in person the same way you would for a first passport. You also have to apply in person if you cannot legally document your name change with a marriage certificate or court order. This process entails visiting a passport acceptance facility or passport agency with your passport photo, evidence of citizenship and identification. There you fill out a DS-11 form, pay your fees in person and wait while they send the application out for you. You'll get your new passport in the mail in about six to eight weeks.
When you don't have time to wait for your expired passport to be renewed or to go through the typical application process, you can expedite the processing. Go to the passport acceptance facility, apply and pay an additional fee for expedited service. If you need to travel outside the country within two weeks, you can make an appointment at a passport agency, where they can furnish you with a passport on the same day -- provided you can prove your need for the urgent processing with documentation such as a plane ticket.