I'll take a crack at this.
Low Fantasy
- Fantasy in which "mundane" tasks, from simple survival to keeping the border around a single town secure, are the order of the day.
- Magic may or may not be common but whatever it can do it cannot solve these "everyday" problems, like finding water or healing grevious wounds.
- Characters are meant to be heroic to the extent that they refuse to give up hope despite apparently overwhelming odds.
- Good examples would be Darksun
High Fantasy
- Fantasy in which grand tasks, international politics or the fufilment of ancient prophecy, take center stage. Averting or winning wars, averting doomsday plots and seting right the wrongs of the age
- Magic is usually both common and reliable. At best it can change the structure of an entire sociery but even at worst it will allow the PC's (if nobody else) to bypass many mundane concerns such as keeping stores of fresh food or having to travel long distances.
- Events usually revlove around organizations (weither they be races, churches, kingdoms or private groups and cults) as opposed to straight up "good vs evil"
- Forgotten Realms and Planescape would be iconic examples.
Epic Fantasy
- May use High or Low fantasy as a starting point but the important differences are about the attitude of the setting.
- The setting emphasises a straight up good vs evil conflict as its main motivating factor and concern for the PC's. Almost without fail the evil is of the "world threating" sort and the the PC's are of the "farmboy who is actually a destined hero" sort.
- Many DnD campaigns fall into this category.
Swords 'n Sorcery
- Often confused with Low Fantasy however there is no particular need for a world where the PC's strugle for daily life or magic is espically rare.
- Magic is reasonably common but is unreliable or requires a heavy price to work thus making it a tool only the deaprate, foolhardy or wicked employ.
- The adventurers are usually footlose/mercenary types (and frequently warrior or rouge types) who are motivated by gold and glory more than a personal stake in things. Villans are always truly vilanous (human sacrafice etc) but the "good guys" may be distinguished only by shades of grey.
- Conan is probably the iconic setting.
Dark Fantasy
- May use High or Low fantasy as a starting point but once again the important differences are all about attitude.
- The Villans and Heroes frequently use the same methods to achieve their goals and, in fact, the similarity of Hero and Villan is one of the central points of the genre.
- Frequently the genre emphasises that, no matter what the PC's do, the outcome will inevitably be bad.
- Focuses on mystical powers that are opposed to humanity: dark fey, demons, undead.
Steampunk
- Has much in common with a High Fantasy setting but replaces much (or all) of the magic with anachrocanistic pseudo-scientific technologh.
- Focuses on a society that has been transformed by these powerful forces.
- A pseudo-victorian setting is common but not required.
In addition we might also add
"Modern" Fantasy
- Fantasy elements are incopropated in a modern or near future world. Elves with machine guns, etc.
- The Urban Arcana setting for Modern.
Science Fantasy
- Sci-Fi traditionally eschews "mystical" elements (except maybe psionics) but Science Fantasy re-introduces these issues. ie: the Force in Star Wars.
Later.
