Air files are so-called Adobe AIR installation files. AIR stands for Adobe Integrated Runtime and is a platform-independent runtime environment for the creation of so-called Rich Internet Applications, ie Internet-based applications for the desktop.
To use AIR applications on a system, you must have a runtime environment similar to Java. These get them for free at Adobe. In addition to Windows, Linux and Mac OSX, Adobe now supports Android, Blackberry and iOS.
An AIR application file contained all the files needed for the application and can run on any supported operating system. The application behaves the same everywhere.