Apple first introduced AirDrop in Lion, making it really easy to file share between two Macs running Lion. In Mountain Lion, Apple have gone one step forward and made this feature available anywhere on the Finder.
In addition to that, Apple also added to the Share button the ability to Email, Message, Twitter, Facebook or Flickr from any where within the Finder (this will depend on what accounts you currently have setup on your System Preferences > Mail,Contacts and Calendars.)
The Share button icon is available on a variety of places throughout OS X Mountain Lion, like Quick Look panels, to Open File dialogs, to Contacts, Safari and Photo Booth.
By clicking on the Share button it opens opens a menu displaying the sharing options. If you open a video those options will change and support for Vimeo and YouTube will be added.
When you select AirDrop you are presented with a dialog that searches for nearby Macs with an AirDrop window open in the Finder.
The share options are different depending on what application you using it on.
As an example:
-> Photo Booth allows you to select a snapshot and perform typical graphic sharing options as well as options to Add to iPhoto, Set Buddy Picture, Set Account Picture, or Change Twitter Profile Picture.
-> Safari, offers to ‘Add to Reading List’ or ‘Add Bookmark’, as well as Email, Message or share via Twitter.