Epic/High Fantasy:
These books are grand in scope. They generally involve a prolonged struggle to overcome some great evil. The success of this battle affects the fate of that world.  Usually the hero/heroine embarks on some quest or a series of tasks that are of high significance. They are generally set in an imaginary world imbued with magic. He/she will have companions who add special gifts/powers/abilities to the mix. The hero/heroine grows from innocence to mastery during the course of the story, often giving the story a coming of age element.
Books in this categories are not just stories. The author creates a world complete with history, language, culture, people, geography and the story emerges from within. This is what gives them such depth and richness.
The author who epitomizes this category is J. R. R. Tolkien in The Lord of the Rings.
Faerie Realm:
These stories feature fairies, elves, pixies, nymphs, dryads and so on. They will either take place in the other realm or in both human and faerie. Or they could take place where faerie coexists with the human world and only certain people can see it as in Wicked Lovely by Melissa Marr.
Fairy Tale:
The books in this category fall into the retelling of fairy tales. Since they tend to have romantic elements and female protagonists, the majority are in Young Adult.
One of my favorites is Phoenix Dance by Dia Calhoun, a unique rendering of The Twelve Dancing Princesses tale.
Fantastic Creatures:
These are stories featuring dragons, unicorns, griffins, goblins, trolls, giants and other imaginary creatures.
Gateway/Magical Objects:
The Gateway part refers to some sort of time travel or just travel to another place either from imaginary to real or real to imaginary. It is almost always paired with the Other Worlds category. There is usually a magical object which triggers the transport to somewhere else. It could be anything: book, tree, gate, clothes, doorway, ball, basement etc. The possibilities are limitless.
The most familiar one of this category is The Chronicles of Narnia by C. S. Lewis.
Myths & Legends:
The most common myths/legends to be used are Greek/Roman, Egyptian, Mermaids, Peter Pan, King Arthur, Robin Hood, and Pirates.
Other Worlds:
Stories set somewhere else, usually some other world or realm. Often paired with Gateway/Magical Object category.
These stories feature ghosts, mediums, angels, vampires and zombies.
Witchcraft & Wizardry:
Here are books about witches, wizards, mages or anyone with magical or special powers.