This style of spell checking is close to Windows 'as-you-type' spell checking. When the inline spell checker is setup (either through Javascript or through Server Controls), the text box will be spell checked either as the user types, or on the click of a button. Underlines will appear and the user has the option to right click on the errors and select a suggestion, Add (to their user dictionary) or Ignore All.
The spell check requests are made through AJAX calls and the server processes them via the RapidSpellWInlineHelper control which sits on an ASP.NET page (by default held in the same folder as the client files).