Where Are AirDrop Files Saved?
In OS X, 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.