The granted power of Tolkien's One Ring is, in fact, to transport its wearer to the wraith world, making the wearer appear invisible from the real world. The wraith world is a parrallel existence inhabited by ancient beings, mainly the elves, Sauron, and other nasties. This is not unlike the Shadowlands of Wraith: the Oblivion, oddly enough.
This is depicted in the film with computerized blurring effect and showing dark landscapes in the background.
source: http://glyphweb.com/arda. As always, their source is mine.