In general, a port of entry (POE) is a place where one may lawfully enter a country. It typically has a staff of people who check passports and visas and inspect luggage to assure that contraband is not imported. International airports are usually ports of entry, as are road and rail crossings on a land border. Seaports can be used as ports of entry only if a dedicated customs presence is posted there. The choice of whether to become a port of entry is up to the civil authority controlling the port.
An airport of entry (AOE) is an airport that provides customs and immigration services for incoming flights. These services allow the airport to serve as an initial port of entry for foreign visitors arriving in a country.
The word "international" in an airport's name usually means that it is an airport of entry, but many airports of entry do not use it. Airports of entry can range from large urban airports with heavy scheduled passenger service, like John F. Kennedy International Airport, to small rural airports serving general aviation exclusively. Often, smaller airports of entry are located near an existing port of entry such as a bridge or seaport.
On the other hand, however, some "former" airports of entry chose to leave their name with the word "international" in it, even though they no longer serve international flights. One example is Osaka International Airport. Even when it had ended all international services and became a purely domestic airport after the opening of Kansai International Airport in 1994, it kept its original name of "Osaka International Airport". Many airports in the nearby region have the same situation, like Taipei Songshan Airport. Songshan retained its official Chinese name, Taipei International Airport, after Chiang Kai-shek International Airport (now Taiwan Taoyuan International Airport) opened. Similar cases of transitions of international airports such as Seoul, Tokyo, Nagoya, Shanghai, Hong Kong, Bangkok, Tehran, etc.