FAR Part 61.60
The FAA, in Federal Aviation Regulation part 61.60, requires that pilots report a change of permanent mailing address within 30 days of moving. Note that this only applies to permanent address changes: a temporary move for college or travel does not count. A pilot cannot use the privileges of his certificate unless he complies with this regulation.
FAR Part 65.21
FAR part 65.21 is a similar regulation that applies to airman other than flight crew members, such as air traffic controllers and aircraft mechanics.
Methods
According to FAR part 61.60 and FAR part 65.21, airmen must notify the FAA of the change via certified mail by mailing a change of address form to the FAA. Airmen can download this form from the FAA's website.
Federal Aviation Administration - Airman Certification Branch
P.O. Box 25082
Oklahoma City, OK 73125
faa.gov
An airman may also comply with 61.60 or 65.21 by filling out the online change of address form on the FAA's website. Before filling out the form, users must first create an online account for the FAA's website. The address will be updated instantly.