Best way to break this down is like this:
Black Metal - Mayham, Immortal and Emperor (Not for ever one)
Doom Metal - Don't really listen to it so can't really name bands
Thrash Metal - Don't really listen to bands from this genre.
Death Metal - Nile, Vader, Death (like black metal not forever one)
Meldoic Death Metal - Dark Tranquillity, Older In Flames, Amon Amarth
Folk Metal - Amorphis, Equilibrium, and Battlelore
Gothic Metal - Tristania, Paradise Lost, My Dying Bride
Power Metal - Kamelot, Firewind, Gamma Ray
Avant-garde metal (aka experimental metal) - I don't really listen to it.
Prog Metal - Dream Theater, Rush, Symphony X (Though you can at time throw Symphony X in with Power metal)
Symphonic Metal - Therion, Nightwish, Delain, Epica, etc (You can in a sense call this the female Power and Prog metal genre Power and Prog metal had a really big hand in make this genre. In fact a lot of male fronted power metal and prog metal bands have been using more and more symphonic elements in there music.
Those for the most part are the key main genres of metal and you can mix and match things with in them
You now have the umbrella terms
Extreme Metal, Female Fronted Metal, Viking Metal, Neo-classical metal, Technical Metal, Christian Metal, Speed Metal.
Most of those any metal band can be or do.
You know have the cross genres In other words it's not really metal and it's really the other things it's a mix
Metalcore, Crust punk, Grindcore, etc. They are either one or the other. Some bands that fall into this don't always do a great job at mix the genres it is either to much on the metal side of to much on the other side of what ever genre they are playing. There are very few metalcore bands that get it right.
Then you have genres that have nothing to do with metal at all
Deathcore, Post-Hardcore, Hardrock, Hardcore (ie Hardcore Punk), etc.
That's kind of how it breaks down.