Fantasy is about making the impossible possible, and even commonplace, in a world that doesn’t live by our rules. It is the ultimate escapism genre.
But fantasy is a large genre. If you’re writing or planning to write a fantasy novel, and you plan to submit to publishers, they’re going to want to know where exactly your book fits in the market. To provide that answer, you need to know the subgenres.

Arthurian

When most people think classic fantasy, they think medieval … and British. (They also usually think Tolkien, but that’s beside the point here.) So, it’s not surprising that there’s a whole subgenre dedicated to one of the most iconic medieval British legends—King Arthur and the Knights of the Round Table. It’s got magic, sword fights, quests, chivalry, and forbidden romance. Plenty of fodder for reimaginings and retellings. These days, it’s become pretty played out, but hey, you could be the one to revive it.

Contemporary (Urban) Fantasy

This is just about the hottest thing on the market right now, especially on YA shelves. Think Cassandra Clare’s interlocking Mortal Instruments, Infernal Devices, and Dark Artifices series (and there are two more trilogies to come; pardon me while I squeal in excitement).
Fantasy elements + modern setting = contemporary fantasy.
Incorporating any feature of classic fantasy, like magic of any kind, fairies, elves, or mystical monsters into your own backyard will work. Typically, “urban” fantasy takes place in urban, big city settings, but that’s not a hard and fast rule. You want a dragon to terrorize a small rural town? You want Fae to snatch away high schoolers in suburbia? Write to your heart’s content. That’s why the more official term is contemporary fantasy, not urban.

Dark Fantasy

This subgenre rides a fine line between horror and fantasy. Many people often use the terms Dark Fantasy and Paranormal Fantasy interchangeably because you’re usually dealing with classic monster-types, like vampires, werewolves, demons, witches, etc. Those creatures/character-types are typically referred to as paranormal.
The biggest identifiable difference between Dark Fantasy and Supernatural Horror is the violence levels. If the story is extremely graphic and violent, it’s Supernatural or Occult Horror. If the story contains violence, but does not give it center stage (think PG-13), it’s Dark Fantasy.
I’d also say that in Dark Fantasy, there’s a huge trend of making the paranormal characters the protagonists, whereas in Supernatural Horror, they’re pretty much always the antagonists. Though there are always exceptions.

Fantastic Alternate History

This is historical fiction with magic thrown into the mix. You usually have a real-life historical figure involved (not necessarily as a main character), but you’re retelling that figure’s story (or the story of their era) with magical elements. As a rule of thumb, Fantastic Alternate History takes place in eras and regions where magic was not accepted as real. So no Middle Ages. No Salem Witch Trials. Think more like Ben Franklin shooting lightning bolts from his finger tips, or a British orphan boy from the early 1800’s who discovers he can perform spells in the vein of Harry Potter.

High Fantasy

This is the ultimate classic. We’re talking Lord of the Rings classic here. The fate of an entire race or nation hangs in the balance, and it’s up to a band of heroes to put things right. There’s usually a quest involved. A massive final battle. Most often, the threat is an individual of unusual or incredible power who has rallied (or is rallying) an army with the desire to reshape the world in his/her twisted image. Without an extremely powerful villain and a huge enemy army, it’s hard to put the whole world at stake.
You’ll typically find the classic fantasy elements here like the dominance of magic, creatures such as elves and dwarfs, medieval flare, sword fights, etc.,  but it’s not actually required. Some element of unreality must be present, or else its not actually fantasy, but as long as you have some fantastical element and the fate of the world (or a huge chunk of it) at stake, you’ve got high fantasy.

Heroic Fantasy

Depending on who you talk to, this subgenre may also be known as Sword and Sorcery, or it may be considered as an overlapping genre of the Sword and Sorcery subgenre, with very subtle differences (so subtle that I myself don’t really see a difference at all and am firmly in the camp of “they’re interchangeable”). Either way, the main characteristic is that it centers on a single hero or heroine as he or she seeks to overcome hardship in an exciting adventure of magic and swordplay. It typically centers around war.
The central hero usually has humble beginnings or some sort of weakness that must be overcome. What generally sets it apart from High Fantasy is that the battles waged are personal to the character and/or the character’s kingdom, and do not influence the fates of an entire race or nation. Think Conan the Barbarian.

Juvenile/Young Adult

Juvenile and Young Adult Fantasy can be written in any of the other subgenres. What set them apart are, first and foremost, the age range of the protagonists. Typically, in Juvenile, protagonists are 8-12 years old, and in YA, 12-18. They are also typically shorter, especially juvenile novels or the first book in a YA series. Think about the Harry Potter books. The first three books are significantly shorter than the last four. That’s because the readers in the target market are growing up with Harry. The first books are shorter and more digestible, and as Rowling moves the series firmly into YA rather than riding the Juvenile/YA line, the books become larger.
Violence, or more like the lack thereof, is another key characteristic. This is far more important in Juvenile, where you want to lean more toward bruises and cuts, and far away from blood and guts. In YA, the boundary of what’s acceptable is growing wider. Over the last decade or so, YA books have started to broach darker topics and include far more violence (even outside the fantasy genre). And they certainly don’t shy away from death. However, as a general rule of thumb, if you’re dealing with violent fantasy battles in YA, don’t linger on the gore. Heroes and villains can be run through with swords, blasted with nasty magic spells, or get arrows through their eyes, but don’t go into detail about what comes out of those wounds, if you get my drift.

Is That Really All of Them?

No. There are a number of smaller subgenres and crossover genres within fantasy. But, they either straddle a fine line with another genre, don’t apply to just fantasy (such as the epic), or are so minute in deviation from another subgenre that it would be difficult to extract one from the other (similar to Heroic Fantasy and Sword and Sorcery).