Hello. I was reading some posts by Jolly Rancher. You've got it mostly right.. But let me continue on a deeper scale.
Nu Metal - A genre of metal that uses negative lyrics, and is usually crossed with rap or grunge. Some good examples are Slipknot, Mudvayne, and Linkin Park.
Black Metal - A genre of metal that usually originates from Norway. It uses an extremely fast tempo, distorted guitar riffs, and often times the vocals sound like high-pitch shrieks. Hey, it's always freezing cold in Norway.. Why wouldnt you create a black metal band? Some examples are Cradle of Filth and Emperor.
Death Metal - A genre of metal that also uses heavily distorted guitars. But unlike black metal, they use very deep screams and growls. Death metal is known to use blast beat drumming, which is a high paced series of ons & offs. Some examples are Behemoth, Disarmonia Mundi, and Suicide Silence.
Deathcore - A genre of music that crosses death metal with metalcore. We will talk about metalcore later. Deathcore uses some deep growls from death metal, yet also incorporates yells and screams from metalcore. Palm-muted riffing is very common in deathcore, and they are well known for their intense breakdowns. Not too many guitar solos. Some examples are Lamb of God, Neaera, and Whitechapel.
Emocore - Yes, emocore is an actual genre of music. They are much like Post-Hardcore music, but use confessional lyrics, and lighter guitar riffs. A lot of falsetto singing is involved. I'd say this genre borderlines with rock. Some examples are Paddock Park and Underoath.
Hardcore - This is the genre that most bands build off of. They simply replace the "hard" with "death", "emo", or even "metal". This is the new age genre of metal that really started it all (other than Thrash). Hardcore uses a fast tempo, and originally evolved from Punk. It is heavier, richer, and uses harsher vocals than Punk. Some examples are Hatebreed, Converge, and The Chariot.
Industrial - Industrial metal is not as common as the other genres. This genre uses traditionally heavy guitar riffs, and harsh vocals. But they incorporate synthesizer. And also, their tempo is faster. Not as fast as Black Metal, but still, quite fast. Some examples are Ministry and Static-X.
Metalcore - Ah. This genre is by far the best in my opinion. Originating in the 1990s, this band is a cross of Heavy / Extreme metal, and Hardcore. It can have both intense guitar solos as well as heavy breakdowns. A lot of the time metalcore brings melody into the picture by having choruses to their songs. Some examples of this are As I Lay Dying, Bullet for my Valentine, Soilwork and Trivium.
Post-Hardcore - Also evolving from the punk scene, Post-Hardcore is a cross between metalcore, punk, and experimental rock. They usually have very loud bass lines, and there is no definition for lyric style. All sorts of lyrics are used. Post-Hardcore is usually half screaming, half falsetto singing. Some examples of this are A Day to Remember, Silverstein, and Story of the Year.
Power Metal - Long hair is a must. This genre of music combines speed metal with heavy metal. They often incorporate symphonic context, and almost all Power Metal bands have a keyboardist. Lots of Power Metal bands sing about medieval times. Some examples are Dragonforce, Nightwish, and Blind Guardian.
Thrash Metal - Last but not least. This is the genre that steered away from traditional "Heavy" metal. This genre started it all. Bands such as Pantera, Slayer and Anthrax. Same heavy metal style, yet they experimented with harsh vocals, and a faster tempo.
Thanks for reading