Classic fantasy is pseudo-medieval, has weird races and magic. Of course fantasy covers a hell of a lot more than that; anything outside of reality can fit in this genre. Fantasy has more subgenres than any other, it seems, if you want to get into labeling things.
My fantasy has a distinct lack of fireballs or, indeed, any magic whatsoever. The idea of a wizard gathering his intrinsic power, or sucking it out of the earth or whatever, and shooting a blazing ball of death at some big baddie leaves me cold. It seems hokey somehow. Even if the magic has consequences and cost (which it had better!), it still seems a bit like a copout to me. Maybe I’m just more interested in the things that make people (or elves, dwarves, efreets, etc) people.
Gandalf… riding out on his white charger… blasting the evil black nazgul dragons with his death ray of purity… once and then he has to resort to whacking things with his staff the rest of the trilogy. Pfft.