As of 2015, a U.S. passport renewal can be started online by filling out the application on the Department of State website. After filling out the application, you must print and mail in the form along with your old passport.
To renew a passport by mail, there are certain criteria that must be met. Your most recent passport must not be damaged in any way. Additionally, you must have been at least 16 years old when the passport was issued, and it must have been issued within the last 15 years. The passport also must have been issued in your current name, or you must have legal documentation showing a name change.