I'd say that, at its most basic form, it is a story that has elements that are completely unrealistic in it-if someone wrote a story in this world where there were aliens I'd call that a fantasy element even though it'd be called sci-fi.
But as a genre I think it's a lot harder to define--when I think of fantasy I usually think of a Otherworld in which there are Othercreatures and Otherraces and Otherbeings--but John Flanagan's fantasy world (in which the Ranger's Apprentice and Brotherband Chronicles are set) is basically an alternate Earth very low on alternate creatures/beings, and there are some books that obviously have fantasy elements, like Ted Dekkers Circle Trilogy or whatever it's called--that is set on Earth.
Again I'd say that Fantasy, at its base, is a genre that includes (or, rather, is based upon a setting, principle, or has a important element that is) something that we know is not real and never will be.
*waits for smarter people to come and give awesome answers*
Great topic idea, Nicole.
