The Tiffany head was designed by Louis Tiffany (yes, that Tiffany). At the time Tiffany developed the Tiffany head, other jewelers were surrounding diamonds with more metal to hold the gem in place. Often, the diamond became overwhelmed by the boxed in type settings.
The design exposes the diamond to greater light from the side and the setting creates space and romances the beauty of the diamond. It was a major innovation for its time and is still popular today for its simplicity and beauty.