Fantasy, or fantastic fiction, is any kind of fiction with fantastic (not realistic) elements. It's worth noting fantasy is not based on reason or rationality. There should be some element of irrationality, such as magic.
Epic fantasy is sub-genre of fantasy with some element of epic-ness: setting, plot, or similar. Often it involves the whole good versus evil battle. Often there's a quest. High fantasy is differentiated from this by focusing a little more on character than on plot. Often there's one main protagonist. Both epic and high fantasy usually involve a secondary world--an imaginary world. Not our modern planet.
Fairy tales, then, have many similar elements of epic and high fantasy. Fairy tales are also called magic tales or wonder tales and are based in myth and/or folk tales. The main thing that differentiates fairy tales from other fantasy is the element of fable, namely, a moral lesson. There's a huge literary trove of fairy tales from Cinderella, to the Little Mermaid, Snow White, Ali Baba and the Forty Thieves, etc. Consequently, there have been many fun modern riffs on all aspects of these tales.
Consider sending us your fairy tale before the upcoming July 15, 2019 submission deadline for the awesome Auguest 2019 issue of Electric Spec.