APO is the official United States Postal Service abbreviation for the Army Post Office. This is an official US Post Office for those in the US military, who may be in unusual parts of the world, and may not have a permanent address.
APO is the equivalent of the city in your address; the state is filled in normally. You can look up state abbreviations, including special military ones, in the United States Postal Service official abbreviations node.
This is one of the few times when you should trust E2 more than the 'authorities'. When you try to send something to an APO address, it's about even odds that your local Post Office clerk will tell you that you have an invalid address. This is not true. It will be up to you to educate them.
See also FPO (Fleet Post Office).