Dark fantasy is more gothic. Vampires, Cthulu… that sort of stuff. It has a definite “horror” quality to it.
High Fantasy is fantasy set in an internally consistent parallel universe like Middle Earth or the Forgotten Realms.
“Swords and horses” I’ve never heard of, but it’s probably referring to standard “Swords and Sorcery” fare, minus all the sorcery.
As far as other subgenres, there are several. Probably too many. Much like with music, it’s always seemed to me that subgenres are created to delineate between things that don’t really need delineation just to say they’re “new” or “different”.
The term “high fantasy” is tricky to pin down, because it’s used to refer to (at least) two different things that are often – but not always – found in the same stories:
A “fantasy world” setting, as opposed to a version of the real world that contains magical elements
An epic struggle against the forces of evil
The contrasting term “low fantasy” is not as common as “high fantasy”, partially because it sounds derogatory even when it isn’t intended to be, and partially because it’s even less rigidly defined. Depending on context it could be intended as a neutral description of either a fantasy novel set in the “real world”* or a fantasy novel that doesn’t deal with epic themes…or it could be used dismissively to describe a fantasy novel that’s considered “low-brow”, especially one that uses a lot of silly humor.
*The term “urban fantasy” has been used to describe stories set in cities in a version of the real world that includes magical elements. I remember this as being a pretty popular subgenre in the '90s. However, it’s my impression that within the past decade or so this term has come to be used more narrowly to refer to romances in which some of the characters are vampires, werewolves, witches, etc.
I may have made that up. It’s just what I call fantasy in a medieval setting, where people are riding horses and swinging swords. “Sword & Sorcery” would fit, too, although I prefer my fantasy light on sorcery - like Abercrombie’s work, or George R. R. Martin’s Song of Ice and Fire series.
“Sword and Sorcery” always makes me think of Marion Zimmer Bradley’s Darkover books,.
And of course that leaves a lot of room for ambiguity. Order of the Stick, for instance, is an epic struggle versus evil in a complete world not our own, but it’s also full of low-brow silly humor. Or then there’s Discworld, which usually doesn’t have the epic struggle, either.
I just think that the Urban Fantasy niche has been swamped with the romances, which I believe are sometimes called “vampire shaggers”. De Lint’s stuff, and the Dresden Files, are still considered Urban Fantasy–they are just outnumbered (right now).
I disagree with the people defining “high fantasy” as “a big story set in a secondary world with a long history, with maps.” I say it is that, but also includes “a struggle between Good and Evil.”
“Dark Fantasy” I would use to describe stories that are “struggles between Evil and Eviller” (like the aforementioned Abercrombie).
“Light Fantasy” is deliberately and primarily humorous, and so includes Order of the Stick and Discworld (Discworld, of course, is also Guilty of Literature, and so is not “just” humorous. Now that I think of it, so is OotS).
The Discworld series is a good example of how muddy the high/low fantasy distinction can be – it’s very explicitly set in a different, magic-based world, but the heroes are only occasionally Saving The World From Evil. Saving This Community From A Bad Ruler comes up a lot more often, as well as other local-level problems. And there is of course plenty of humor!
Conversely, the Wikipedia article on high fantasy uses the Harry Potter series as an example despite the fact that the series takes place in a version of the real world that includes magic. The Wiki article claims that since Hogwarts “is physically separated from the real world” it “is therefore as much of an alternative world as C. S. Lewis’ Narnia, which means that both series are in the high fantasy subgenre”, but this reaaaaaally seems like a reach to me. Harry Potter is high fantasy in the sense that it does eventually come down to a struggle in which Harry & Friends must Save The World From Evil, though.