Adobe air is also known as Adobe Integrated Run time. used to run Flash based and RIA based Desktop clients.
You might be wondering why we need AIR, even though we have Flash Player Installed in our machine.
Actually Adobe Provided a Software called Adobe Flex Builder to build RIA very fast. The core libraries are Flash Libraries only. But on top of those libraries they built components, charts to Build flash websites very easily. To run these websites we need Flash Player.
But there is also an option for creating desktop applications, where they can be installed on to your machine and run as normal windows applications. The good thing here is whatever you can do with flash, you can do on this windows. These applications are called AIR/Desktop/Offline Applications.
To run these applications we need AIR run time. It's a super set of Flash player libraries plus some Other libraries.
I hope this Helped you!