The AIR app won't work in a browser. For the web version, you would use Flex and for the desktop version you would use AIR. Both applications can be written in Actionscript 3 however, unlike Flex apps, AIR apps can also be written in HTML/JS. If you write your apps in Actionscript 3, your Flex and AIR apps could use almost identical code. Obviously, there are a few classes which can only be used in AIR as these related to desktop functionality. If you look at the Actionscript 3 reference (http://help.adobe.com/en_US/FlashPlatform/reference/actionscript/3/), you can see which of the classes are AIR only and which can be used in both Flex and AIR.
AIR files are exported as .air files while Flex file are exported as regular Flash .swf files.
Thanks for breaking it in to details, its a good information to make the choice as a new person (coming from pascal/c/c++/c#), reading 500 pages only to make choice for air/flex/flash seems crazy to me. Thanks a lot.