I honestly can't give you definitions to each, just good examples to compare each to. I don't know enough about metal to be able to scientifically explain what composes each genre, you'll have to wait for Helm for that.
Thrash: Metallica, Megadeth, Slayer, Sepultura, and Anthrax. Sepultura andMetallica's first few albums and Anthrax pre-Public Enemy stuff would be considered thrash, but none of their latter work. See also: S.O.D., M.O.D., Exodus, Nuclear Assault, Psychotic Waltz (if you can obtain something of good sound quality), Sentenced, and Sodom.
Death: Death, Morbid Angel, Deicide and Obituary are the Florida Four of this style. See also: Dismember, Entombed, Posessed, Brujeria, Cannibal Corpse, Blood, Exhumed, Pungent Stench, Autopsy, Bloodbath, and Nile.
Black: Black Metal's not really my forté, so i don't know anything about its founding fathers. Good bands that i know of include In The Woods..., ...And Oceans, Arcturus, Burzum, Antaeus, Anaal Nathrakh, and Angel Corpse. I've been told that At the Gates are Death and Melodic Death and all that shit, but i think they remind me more of BM than anything else. One of those genre-benders, methinks.
Doom: Traces of Sabbath can be seen in all of these bands. Sabbath and Pentagram might be seen as two founders of this genre. Also try Candlemass, My Dying Bride, Anathema, Bongzilla, Sloth, Cathedral, Eyehategod and Iron Monkey.
Grind: Carcass and Napalm Death are revered on this scene, with Terrorizer and Agathocles also being oldsies. New bands include Anal Cunt, Gerogerigegege, Nasum, Bolt Thrower, Pig Destroyer, Brutal Truth, Asshole Parade, Cattle Decapitation, Agoraphobic Nosebleed, and Phobia.
Power: Don't know about the founders of this either, but it's like modern Glam Metal. Check out Rhapsody, Sonata Arctica, Stratovarius, Avantasia, and Manowar.
Sludge: See
Doom.
Gore: See
Death, add
very descriptive lyrics about killing people.
Industrial: I don't know much about it, nor do i care to know. Fear Factory are the only band in this genre that i know.
Prog Metal: I avoid it.
Noise: Another word for
Grind. They're the same thing.
Heavy Metal: Old shit with clean vocals. Judas Priest and Iron Maiden are the only ones i can think of. I don't care for either of them.
Hair: All the cheesy 80's shit. Twisted Sister and Manowar are the shining examples here.