-Progressive is just an adjective that can apply to any rock/metal, it simply means that the musicians usually have more complex song structures, as well as including influences from stuff like jazz. Usually songs are longer too.
-Death Metal is a pretty broad genre since there's a lot of variations of it. The genre came from thrash. Bands I would simply call "Death Metal" would be Morbid Angel, Cannibal Corpse, Deicide, Immolation, stuff like that. Songs are usually fast but not too fast, with regular leads and growled/guttural vocals.
example:
YouTube - Morbid Angel - Fall From Grace
-There is also Brutal Death Metal, which is usually faster and...well more brutal than what I would call just "death metal". Lots of blast beats, deeper, more guttural vocals (not always), heavy ass riffing. Bands in this genre include Suffocation, Disgorge, Defeated Sanity.
example:
YouTube - Suffocation - Effigy Of The Forgotten
-Then there is technical death metal. Tech Death is like brutal death with more leads and guitar wanking...it's just more technical. I'd say tech death started in the mid 90s. A lot of bands will fall in both genres. Cryptopsy, Origin, Gorguts, Spawn of Possession and Necrophagist are some examples of artists.
example:
YouTube - Origin - The Aftermath
-Thrash is the original extreme metal, it was really popular in the late 80s. The vocals are usually higher and not growled. There's a lot of shrieking guitar solos and fast riffing, usually they don't tune down as low as death metal...thrash is like early Metallica, Slayer, Kreator.
example:
YouTube - Kreator - Awakening of the Gods
-Grindcore is hard to describe...it's fast, but not really technical. Honestly I don't listen to a lot of it with the exception of the standout artists. I'm going to include Deathgrind and Goregrind in here too cause I'm lazy. Deathgrind and Goregrind are like Grindcore and Death metal combined, Deathgrind is usually more complex...Goregrind is usually really fucked up subjects about gore and rape and shit. The guitars sound like shit cause they're suppose to. The vocals are pretty diverse usually screaming or growling. Artists include (early) Carcass, Agoraphobic Nosebleed, Napalm Death.
Grindcore/Goregrind ex:
YouTube - Carcass - Reek of Putrefaction
Deathgrind ex:
YouTube - Cephalic Carnage - Lucid Interval
-Metalcore and Deathcore are pretty straight forward. Metalcore is like a metal'd up version of hardcore punk with a lot of breakdowns. Deathcore is metalcore with death metal and more "bree bree" vocals. Metalcore bands are like Ion Dissonance and Atreyu. Deathcore bands are stuff like Suicide Silence, All Shall Perish, and As Blood Runs Black.
Metalcore ex:
YouTube - Trivium - Dying In Your Arms
Deathcore ex:
YouTube - All Shall Perish - There is no Business to be Done On A Dead
-There's also black metal, which pretty much came at the same time as thrash. And blackened death metal which is mixing death and black metal. Ya'll can research those on your own cause I just wrote a shitload about extreme metal...