As a self-professed genre-bender, who likes to mix and match fun genre conventions regardless of origin, you'd think I wouldn't care much about the rather nitpicky and esoteric subgenres of fantasy, but actually, I think that they're quite interesting to examine. As a reader, occasional writer, and long-time fantasy roleplaying game fan, I also care quite a bit about the "feel" of certain genres, and often try to implement this feel regardless of other conventions sometimes.
This list isn't meant to be exhaustive, although it does catch most of the major categories.
High Fantasy: The grand-daddy of them all. Well, not really, but High Fantasy is what so many people think of when they hear the term fantasy that for all intents and purposes, to them High Fantasy is synonomous with Fantasy. High fantasy usually takes place in a secondary creation (i.e., a fantasy world, not the real world), often includes quests to save the world, often includes a dynamic of good vs. evil, and often follows such conventions as Campbell's Hero's Journey very blatantly, including the farmboy turned hero/king. Tolkien's Lord of the Rings is High Fantasy, for example.
Sword & Sorcery: This is the other major pillar on which most modern fantasy rests, and it was pioneered by writers such as Robert E. Howard, Clark Ashton Smith, Fritz Leiber, etc. S&S does not feature "save the world" as a plot element, it is often told in short story format, the protagonists are often fairly amoral guys, looking for adventure and glory rather than higher ideals. It's focus is on exotica, action, and fast pacing. S&S originated in the pulp magazines (especially Weird Tales of the late 20s and early 30s) and as such, it bears a strong helping of the "pulp aesthetic." As already mentioned, the most prominent and iconic Sword & Sorcery fiction is the Conan cycle of stories by Howard.
Heroic Fantasy: This is a slightly unclear definition, and seems to be used most often to label something that would include both High Fantasy and Sword & Sorcery, making it an umbrella term of dubious value. However, it's worth pointing out that games like Dungeons & Dragons, which feature an intriguing mix of High Fantasy and Sword & Sorcery conventions mixed together, is perhaps best served by this label.
Low Fantasy: This is probably a bad label, but you may see it frequently. It's defined more by what it's not than by what it is, i.e., it's not High Fantasy. So, this includes Sword & Sorcery, for example. And dark fantasy
Dark Fantasy: Best described as a hybrid between horror and fantasy. Some dark fantasy "leans" more towards one endpoint or the other. Dark fantasy is also often "non-heroic" i.e., characters are vulnerable to nasty death at every turn, often don't have plot immunity, and the grittiness and "realness" of life is emphasized. Glen Cook's Black Company series could be considered dark fantasy (as well as other things), as could George R. R. Martin's Song of Fire & Ice.
Contemporary Fantasy: Fantasy that doesn't take place in a pseudo-medieval setting. It could be, literally, the real world of today, or it could be a fantasy setting with modern trappings. The most obvious and popular contemporary fantasy today is J. K. Rowling's Harry Potter series.
Urban Fantasy: A subset of Contemporary Fantasy, urban fantasy takes place in the dark alleys of the modern metropolis. Growing out of a dark fantasy/horror scene, urban fantasy features a lot of overlap with that scene. The Anita Blake and Dresden Files novels are a great example of Urban Fantasy. For that matter, so is much of the World of Darkness line of roleplaying games from White Wolf.
Romantic Fantasy: Fantasy mostly written by women and targeting a women audience, this subgenre combines elements of traditional high fantasy with the romance novel. Mercedes Lackey wrote a ton of this kind of material, and a roleplaying game by Green Ronin dedicated to it even came out, Blue Rose. I don't know how successful that game was, but it was the direct predecessor to the True20 system, so it's worth it for that alone, if nothing else. In case a bit of bias is showing through here, I'll go ahead and officially say it: I have yet to read a single book of this subgenre that I like, and I actively avoid it now.
Science Fantasy: Perhaps not so much thriving subgenre in its own right as opposed to a label retroactively applied to a few works that don't easily fit in other subgenres, science fantasy is a subgenre of fantasy that has science fiction trappings, but other conventions (and even cliches) from high fantasy, or other fantasy works. Frank Herbert's Dune is probably one of the earliest well-known varieties of this, but the Star Wars franchise is usually what it meant when this label is applied. Of course, things like Warhammer 40k could also be called science fantasy, and for that matter, you could easily retroactively call a lot of space opera science fantasy today, like E. E. "Doc" Smith's Lensmen series.
Historical Fantasy: As the name implies, this is fantasy that takes place in a historical setting. Celtic fantasy, Homeric fantasy, Arthurian fantasy, etc. If it takes place in the real world, using real (or real legendary) characters, at least as window dressing, if nothing else, then it's historical fantasy. Regardless of how much magic, monsters, dragons, and whatnot happen to infest this fictional real world.
Epic Fantasy: You probably hear this term a lot, but it is unclear. Some people use it to mean something that's identical to High Fantasy. Others use it to refer to how long a work is. Neither is a particularly useful definition, in my opinion, so I tend to ignore this label as nonviable.