Image via WikipediaNow, if you are using Adobe Flex Builder to build AIR, you can leverage on Facebook API by using the 'ActionScript 3.0 Client Library for Facebook Platform'.
This new library is open source and support for 60 new APIs.
For getting started and videos, check out here.
The main idea of using Facebook API is to connect to more people. For example "Playfish has leveraged Facebook Platform and the Flash Platform to create fun and engaging social games that are now played by more than 60 million registered users around the world," said Sebastien de Halleux, chief operating officer and co-founder of Playfish.