The original purpose of the Featured Image was to serve as a page (or post) specific header in place of the general site header. So as well as being able to customise your site’s header on a site-wide basis, you could use the Featured Image to make you post or page unique.
Not all themes support this type of behaviour any more. The default themes that come with WordPress out of the box seem to have stopped supporting it in this way.
Theme and plugin developers also use it for other purposes in a different context.