If you want to renew your passport by mail, you must send your current passport to the passport office along with a renewal application. This is only temporary, though. Once the passport processing center is finished processing your renewal, they will return your original passport to you with markings so it cannot be used again.
The exceptions to this rule are if your passport has been lost, damaged, or otherwise altered. Under these circumstances, passports cannot be renewed by mail. In order to renew a lost or damaged passport, you must apply in person at a passport office.
Even if you do still hold an undamaged passport in your possession, there are addictional factors you must meet in order to renew through the mail. These include:
Possessing a passport that was issued after the age of 16
Possessing a passport that was issued less than 15 years prior to renewal
Having the same name you had when the original passport was issued
If you meet these three criteria, you can renew your passport by sending it in to the passport processing center with your application. If you fail to meet these criteria, you must reapply for your passport at a passport office.