A nonexistent object is something that does not exist. Some examples often cited are: Zeus, Pegasus, Sherlock Holmes, Vulcan, the perpetual motion machine, the golden mountain, the fountain of youth, the round square, etc. Some important philosophers have thought that the very concept of a nonexistent object is contradictory (Hume) or logically ill-formed (Kant, Frege), while others (Leibniz, Meinong, the Russell of Principles of Mathematics) have embraced it wholeheartedly.
One of the reasons why there are doubts about the concept of a nonexistent object is this: to be able to truly claim of an object that it doesn't exist, it seems that one has to presuppose that it exists, for doesn't a thing have to exist if we are to make a true claim about it? In the face of this puzzling situation, one has to be very careful when accepting or formulating the idea that there are nonexistent objects. It turns out that Kant's view that “exists” is not a “real” predicate and Frege's view, that “exists” is not a predicate of individuals (i.e., a predicate that yields a well-formed sentence if one puts a singular term in front of it), has to be abandoned if one is to accept the claim that there are nonexistent objects.
See full article at http://plato.stanford.edu/entries/nonexistent-objects/