According to the U. S. Department of State (travel.state.gov/passport) one can only renew his or her passport by mail if your most recent passport is undamaged and can be submitted with an application.
Also, If your most recent password was issued to you when you were 16 or older, if your most recent passport was issued within in the last 15 years, and your most current passport was issued in your name or you can account for a legal name change. Note that all these requirements apply otherwise you have to renew your passport in person.
From within the United States(or from within Canada) one can renew a passport by mail by completing Form DS-82 Application for a Passport By Mail (http://travel.state.gov/passport/forms/ds82/ds82_843.html). The form can be completed online or by hand.
Additionally, your most recent passport must be included and 2 passport photos. Each photo must be in color, identical, taken within the last 6 months, be 2" width and 2" high, be full-faced (between 1 inch and 1 3/8 inches from the bottom of the chin to the top of the head), and be in normal street attire (For additional requirements visit http://travel.state.gov/passport/get/first/first_830.html#Photos).
Additionally, current passport fees should be included.(http://travel.state.gov/passport/get/fees/fees_837.html).
Checks and money orders can be used to pay for passports by mail.(http://travel.state.gov/passport/get/fees/fees_837.html#pmt_instruct).
Make sure your current passport, the passport form (if applicable), passport photos and passport payment are sent by certified or traceable mail.
One should visit a U.S. Embassy or U. S. Consulate:(http://www.travel.state.gov/travel/tips/embassies/embassies_1214.html) to renew a passport if one is outside of the United States or Canada.