Fantasy is an umbrella term that contains a wide range of subgenres that can sometimes be a bit tricky to navigate. Therefore, I am being a good little dragon and offer you a list of many of them (I can't say all because new ones seem to pop up all the time). :3 Do keep in mind that many of these overlaps, so you don't get too confused. I've put a * next to the ones who are more common than the rest.
Tales set in the world of King Arthur's legend. Camelot, Merlin, Lancelot, Morgana, and all that jazz. These stories have been popular for centuries and will most likely continue to be so (hell, I like them even though I know they're inaccurate, even the ones who say they're historically accurate. On that note, here's an interesting article on the accuracy and history of the Arthurian legend). It can also be called Medieval Fantasy, featuring kings and knights, dragons and sorcerers and so on, without it having to be related to the King Arthur story.
Pretty self-explanatory. I've found that most fantasy stories in this subgenre are satires and a breakdown of the tropes most of the time, of varying grades of success. Pratchett is one of the more prominent names in this genre and one that does it well.
Lots of YA Fantasy fall under this one, even if they're trying to market themselves as High or Epic. It's set in royal castles, either in an alternative universe or in historical ones, and full of nobility, usually a poor main character, and some love triangle or another.
This can be considered the fantasy genre's answer to horror. The world is bleak and usually dystopian, the themes are dark and frightening (compared to regular fantasy), and can be told from the perspective of a "monster". My experience is that this is the only genre where the main character doesn't win is acceptable.
It centres around either a hero, a band of heroes, or an unwilling hero. The latter is by far the most popular version to write about (see Inheritance Cycle, Lord of the Rings, Black Magician, to name a few). It contains a wide variety of characters, including forgivable villains, and the main character is usually quite flawed (although this is something that is becoming more and more popular in other genres as well, as younger readers find this more interesting to read about. Loki from Thor is probably the most prominent current example). Unlike Sword and Sorcery, this one has a tendency to highlight the personal struggles more than the world-ending ones.
High or Epic*
Stories set in a completely different world, usually epic in nature (hero on a quest of some sort). Often, though not always, contain magic and/or monsters of varying degrees. This is by far considered to be the heart of the genre as a whole and what most think of when you mention "fantasy books". Some stories are majorly set in an alternative world unlike our own, but still has sections in our world (Narnia for example). Some people bunch these together with High/Epic Fantasy, not entirely incorrectly but not correct either. For those types of stories, there is a separate subgenre called Portal Fantasy.
This can be considered the fantasy genre's answer to historical fiction. The setting is some period of Earth's history, but with elements of fantasy added to it. The British TV series Dr Who does this very well.
Unlike High/Epic Fantasy this subgenre contains less magic and has a more ordinary setting, often in our world. Heroism isn't the focus of these stories, and there's usually no quest. It's tightly connected to Urban Fantasy but doesn't' take place in an urban area. Harry Potter is therefore not Urban Fantasy, but Low Fantasy.
Unlike Historical, Mythic stories take place in ancient times. Myths from classical mythologies and folklore are considered real (not necessarily all of them). There are Low Fantasy stories that contain elements of mythological origin, but that doesn't mean the story is actually part of the Mythic subgenre. Stories such as Hercules's legend, and Homer's The Odyssey and The Iliad are also part of the Mythic subgenre. Some do consider stories taking place today with existing myths incorporated to be Mythic instead of Low fantasy, so please be aware that this is my interpretation of the genres.
Sword and Sorcery*
This is quite closely related to Heroic Fantasy, although this one is a more traditional take on the quests and the hero's journey. The world is in the balance, and the chosen one is their only hope. Magic, heroes, swords, thieves, and all the things you stereotypically associate with fantasy is found in this category. It's less royal palaces and more underground and sleeping in the forest than Heroic Fantasy, to describe it quickly.
One of the most popular subgenres for Young Adult writers. It's a fantasy story that takes place in a modern city, plainly put. The Mortal Instruments is an excellent example of this. It's an urban city, with fantastical elements added to it, and has a typical fantasy quest. Some use this term for any story that takes place in our world that has fantastical elements such as vampires and magicians, but that's incorrect. Most stories I've read that contain vampires, for example, takes place in a small town. "Rural Fantasy" is a term I've heard being used for that and I think that's quite fitting, although there is a genre for stories that centre around vampires and such already: Paranormal Fantasy.
Among the most popular subgenres for Young Adult writers. It's a fantasy story that centres around supernatural beings such as vampires and werewolves. Most stories I've read in this subgenre contain elements of romance. The Twilight Saga fits in here.