You download the installer from here (it's called AdobeAIRInstaller.bin and it's hosted on http://get.adobe.com/air/).
Gov to wherever you downloaded it and make sure you have permission to run it:
In Nautilus right click on the file
- choose Permissions
- Check the box that says Allow executing program as a file
In the terminal
- chmod +x AdobeAIRInstaller.bin
Then all you have to do is run the installer program:
Accept the licensing agreement and it should install just fine.
AIR stands for Adobe Integrated Runtime. "Runtime" here means an environment for running applications. Adobe's runtime environment let you run on your desktop applications that you normally run from the web using your web browser.