(Note that I'm excerpting from a reply I made to a previous thread)
Adobe AIR is a desktop runtime -- a "program" that runs on your computer and itself runs other programs. It can be compared to Flash Player for the web browser. You can view plain HTML pages in a browser without needing anything extra, just like you can run native programs on your computer without needing anything extra installed. However, some web sites include Flash content, which only runs in the browser with help from Flash Player. You never "run" Flash Player by yourself -- it is run automatically by the browser, and it runs SWF file content (video, ActionScript code, etc.). Adobe AIR is used to run desktop applications rather than content in the web browser, and just like a SWF file doesn't run in the browser without Flash Player installed, an AIR application won't work on your computer without Adobe AIR installed.
Most people in the world will just use Adobe AIR applications built by developers. If you are in that situation you can just think of AIR as something that your computer needs in order to run their application, just like Flash Player is needed to view certain web sites.