Flash is an application development tool. This application can be anything from a small banner going to a site onclick, a facebook game, a multiplayer 3D RPG, a lab simulator ... you name it!
It has a tremendous library support and it can do actually everything.
Air can export to .ipa (iphone/ipad application) and you can then submit to appstore.
A classic sample of a big hit written in flash is Machinarium and another one is angry birds for facebook.
For more details about Air you can see here http://www.adobe.com/products/air/fe...es_sl_new.html