This post will perhaps be more germane to literature than to gaming, and it will be short (because I haven't the time write anything longer at the moment), but I feel that I have to get it out there. One my personal pet peeves is the misuse of genre labels, and of fantasy sub-genres in particular. I don't presume to speak with authority on sub-genres of science fiction because I'm admittedly not well-read when it comes to classic sci-fi. But I should hope that I know what I'm talking about by now when it comes to fantasy.
There are a number of sub-genres that tend to get mixed up or conflated, the term for one applied to another, and I find this bothersome. Of course the boundaries between genres are never clear-cut, and many stories mix and match genres or belong to multiple genres at once. But it's still helpful to have clear definitions.
I'll begin with the most badly-abused terms: high fantasy (which is not epic fantasy) and low fantasy (which is not sword & sorcery).
High Fantasy: Stories that take place in a fantasy world which is not Earth, also called an invented world or a secondary world. Notably, a pure high fantasy should have no connection to Earth (see portal fantasy) and neither should it be a period in Earth's past (mythic fantasy). That's the only criterion for making a story high fantasy: the setting is another world that isn't ours. Every Final Fantasy game and most D&D campaigns are high fantasy for this reason.
Low Fantasy: Stories that take place in our world (regardless of time period), where the fantastical element is an intrusion on otherwise mundane reality. A story about a child's toy coming to life, or a story about meeting a fairy or a wizard in what otherwise appears to be a real-world setting, is low fantasy. Children's stories and Hollywood movies lend themselves to this genre very well; tabletop games generally do not.
Mythic Fantasy: Now. admittedly. this term isn't in common currency, but I think it's a useful one nevertheless. Given the definitions of high fantasy and low fantasy presented above, are Middle-Earth and Conan's Hyborean Age high or low fantasy? Both explicitly take place not in some alternate universe, but on Earth. (Tolkien famously hated it when people suggested to him that Middle-Earth was "another world.") However, the magic in these stories isn't an intrusion on a mundane world—it's fundamental to the setting, a speculated mythic past when magic was more prevalent in our world than it is now. (Curiously, too, both settings are rather circumspect about whether the magic actually is magic at all—Tolkien's elves are perplexed by the term for what is, to them, a perfectly ordinary aspect of their very nature, and sorcerers in Conan stories have been known to imply that their "supernatural" knowledge is merely an advanced science.) I think it's a good idea to draw the distinction, and say that mythic fantasy is the term we should use for a story that takes place in a mythic or legendary past version of Earth. The Hobbit, Conan the Barbarian, and Xena: Warrior Princess would be prime examples of the genre.
Sword & Sorcery: This is where the genres start to overlap. The previous definitions concerned the setting; this one is about the narrative. Sword & sorcery stories are those in which the focus is on a single hero or band of heroes, and their concerns are personal rather than world-shaking. Such protagonists are nearly always warriors (Conan, Elric, etc.); they often have to deal with magic, magic-users, monsters, or other supernatural threats. But what's key here is that the heroes' goals are personal and often clearly achieved (or not) by the end of a single short, pulpy story. Nobody "saves the world"—the heroes aren't "chosen ones"—and their goals are hardly ever quests.
Epic Fantasy: This is the proper term for that generic sort of tale where a reluctant everyman—possibly a chosen one with a hidden royal heritage or a mysterious, prophesized destiny—gets dragged into an epic quest to save the world. Epic fantasy is all about mighty upper-class knights and kings and wizards (and maybe one protagonist farm-boy) striving to preserve the status quo from an external, world-level threat (the Dark Lord being a perennial favorite).
Heroic Fantasy: This term is less-than-useful, because it's just any fantasy with action, adventure, and a hero. High fantasy, sword & sorcery, and epic fantasy nearly always fall within the broad category of heroic fantasy; low fantasy may or may not, but usually doesn't.
Finally, honorable mention to two more related sub-genres that can overlap with any of the above.
Portal Fantasy: This is the term for tales of characters transported from our world into a fantasy world. Narnia and Oz are the classic examples, but there are countless others. Since the "other world" is by definition a secondary world, portal fantasy is almost always high fantasy as well, just by definition. But, for the love of Pete, don't go calling all portal fantasy "isekai." If you're not dealing with manga, anime, or characters who get transported into an RPG and remain aware of their own stats, isekai is emphatically not involved.
Urban Fantasy: This is the sub-genre that describes a version of our modern world, but where magic exists (either openly or hidden behind "the masquerade"). It is, in effect, mythic fantasy taking place in the present day, or low fantasy with the "fantasy" aspect dialed up to eleven. Harry Potter probably technically qualifies as this genre, although the more traditional example would be something like The Dresden Files.
So, there you have it. The major fantasy sub-genres. (I've probably forgotten or neglected a good many more, but these are the ones that concern me at the moment.) Use them, don't abuse them, mix and match to your heart's delight… just quit calling all sword & sorcery stories "low fantasy," or all epic quests to save the world "high fantasy," because that's annoying.