What is Fantasy ? "Fantasy is the stuff of myth, of flights of fancy, folklore, fairy tales, magic, and heroes."

Some  major subgenres and themes of fantasy are:
Sword & Sorcery
           (Quest, Epic, Heroic, Wizards, Sorcerers, and Enchantresses)                      
Saga, Myth, and Legend
           (Arthurian Legend, Robin Hood and Sherwood Forest, Ancient
           Civilizations, Celtic, The Americas, Asian, European, Africa and the
           Middle East, Byzantium)
Fairy Tales
           (The Familiar, Originals, Fairy Tale Short Stories)
Humorous Fantasy
A Bestiary
           (Anthologies, Unicorns, Dragons, Uncommon Common Animals,
           World of Faerie)
Contemporary
           (Urban Fantasy, The Human Condition, Magic Realism)
Alternate & Parallel Worlds
           (Alternate History, Parallel Worlds, Alternate Worlds)
Time Travel
Paranormal Powers
           (Psychic  Powers, Shapeshifters, Immortality, Supernatural Beings)