In OS X, any file received via AirDrop is automatically placed into your Downloads folder. The Downloads folder can be found inside of your Users folder. Or from the Finder, push Option+Command+L to open it.
In iOS, AirDrop files can be retrieved by opening the app associated with that file. For instance, if you were sent a photo via AirDrop you will find the photo in the Photos app. Videos will go to the Videos app, etc.