High Fantasy, or Epic Fantasy focuses on the world itself and typically has themes of good versus evil. Lord of the Rings is the archetypal example. This is where you'll find your typical Elves, Dwarves, Gnomes, Hobbits, Dragons, Dark Lords, Magical Powers, Wizards, Ancient Prophecies, The Chosen one, etc, etc.
Heroic Fantasy focuses on the exploits of the heroes themselves, and tends to be a bit more low key than High Fantasy. White and Grey/Grey and Grey morality is present, the scale is smaller,the hero's quest tends to be more personal than world ending. Conan the Barbarian is the archetypal example. Another good example are most Dungeons and Dragons campaigns. Sword and Sorcery and Wuxia style stories are typically similar to Heroic Fantasy, although Wuxia High Fantasy also exists.
Low Fantasy focuses on the mundane and gritty. Magic is more subdued, humans are typically the dominant race, and there's a big emphasis of Grey and Grey morality. Game of Thrones is a good example of low fantasy.
Urban Fantasy is fantasy but set in a modern urban setting. Persona is one such example.
Dark Fantasy is fantasy meets horror. Science Fantasy is fantasy meets Sci-Fi
There are countless other such examples and sub-genres that would take forever to explain if we tried. That Wikipedia article above is a good read and your best bet for learning about a specific type.
