Welcome to Gaia! ::

I'm sure someone more qualified can do it then.

i see nothing to add to grave's explanation of thrash.

Black Metal: black metal is typically divided into three "waves" or movements.
First Wave BM is not a true form of black metal, but is merely a convenient way of listing black metal's biggest influences. these include heavy and thrash bands with a darker lyrical and musical focus. the first wave bands are celtic frost, hellhammer, (early) bathory, venom, and mercyful fate.
Second Wave BM developed in the late 1980s and early 1990s. these bands, mostly from norway and sweden, developed a radically new style of playing music, employing a style focusing more on the development of dark/evil atmospheres and less on technical proficiency. typically played at a very fast tempo, but without the influence of hardcore punk brought to thrash. musically simple, with moments of dissonance utilized to build atmosphere, and commonly utilizes blastbeats on the drums. much distortion is placed on the guitars, and because of financial difficulties, many bands recorded low(er) sound-quality releases than typically seen in the metal genre, adding to the dark atmosphere. typically bass is mixed very low/nonexistent, though some bands incorporate it very well into their sound. lyrical focus on satanism/evil, reverence for nature, pagan themes, misanthropy, norse culture, etc. many bands wore corpse paint to distinguish themselves from "untrue" metal bands. some examples of 2nd wave BM bands are gorgoroth, mayhem, (early) emperor, immortal, and darkthrone, among others.
Third Wave BM is harder to define. there is no clear cutting off point between where the 2nd wave ended and the 3rd wave began. generally, however, the 3rd wave is considered to have started with a revival of interest in BM towards the turn of the millennium. 3rd wave bands are generally criticized for being unoriginal - whereas each of the 2nd wave bands sounded very different from the others and developed their own style, 3rd wave bands have a tendency to rip off the sounds of darkthrone. they are also criticized for using deliberately bad recording techniques - in the old days, BM had low sound quality releases because the bands could not afford good recording equipment or professional producers. what made 2nd wave BM so good was that, despite these obstacles, the recordings still effectively produced a dark atmosphere and were enjoyable to listen to. in these days, recording equipment is cheap and a near-professional sounding demo can be recorded on a laptop, and the bands (mostly darkthrone-clones) who use deliberately poor techniques do not further the genre into new heights of artistic progression, but merely encourage it to stagnate. there are, however, a few good 3rd wave black metal bands, such as nachtmystium and deathspell omega.

a couple notes to add to the doom metal section:

Although the foundations of doom were built and developed within Black Sabbath first few albums, it wasn't until the early 80s that those foundations were further rooted into the ground with Witchfinder General's demo Burning A Sinner and album Death Penalty, Pentagram's demos and albums, Force (the band before they were called Iron Man, although not as well known or as influential as their contemporaries, the fact remains it is still there), Trouble, Nemesis/Candlemass, Pagan Altar, and Saint Vitus.

And I'd just like to add that in my experience with epic doom it is more high-flown which you can see and hear in general themes, musicianship, and delivery; having a very clean and clear production is also a constant presence among epic doom. Some notable examples to listen to: Candlemass, Doomsword (all but latest, which is pure epic heavy metal), Solstice, Scald, Heathendom, While Heaven Wept

I feel the need to redo Folk Metal. I'll do it when I come home tonight.