I'll put the description explaining each type of Metal genre as it can be confusing:
- Also called "traditional metal". This is the most "basic", rock-oriented kind of metal. You can expect a lot of guitar solos, high pitched vocals, and catchy, hook-based songs when you're listening to heavy metal. Some of the most famous metal bands play traditional metal, such as Judas Priest, Iron Maiden, Manowar, and King Diamond.
- Doom metal is shockingly hard to pin down. It's debatably the oldest kind of metal, having been invented along with metal itself with Black Sabbath, and the subgenres of it have developed so far from the basic sound that about the only thing you can say about it is that it's slow and heavy. Traditional doom metal bands such as Black Sabbath and Witchfinder General are quite popular, but any metalhead needs to be aware of the representatives of death/doom and funeral doom styles - look for bands such as Evoken, Catacombs, My Dying Bride, and Autopsy.
- The average listener is probably more familiar with thrash metal than any other genre, as Metallica, Megadeth, Slayer, Anthrax, Testament, and many other extremely well known bands either fall under its umbrella now, or did at some point in their career. Thrash metal is always influenced by punk rock and hardcore to some degree. Common elements include fast tempos and palm-muted, E-string based riffs. The vocals can be either shouted or sung.
- Mainly an evolution of thrash metal. Death metal actually started as just heavier, less melodic thrash metal bands, usually featuring downtuned guitars - check out old albums by Possessed, Death, and Morbid Angel for some good examples of this. These days, death metal has plenty of variety, but the core elements are still amelodic riffing, growled or screamed vocals, and tempos that are either very fast or at least very erratic.
- I regard black metal as being aesthetically pretty different from the other kinds of metal, and also as one of the hardest genres to pin down in a description. Most black metal nowadays follows the blueprint developed by Norway in the early 90s, meaning look for grainy production, melodic, tremolo picked riffs(tremolo picking is just picking one note over and over very quickly), and "blast beat"-based drumming. Early advocates of this style include Darkthrone, Mayhem, and Emperor. However, like the other genres, there's no shortage of variety.
- "Progressive" is a term used to describe both rock and metal which feature uncommon, even experimental, elements, frequently taking the form of a heightened level of complexity and an emphasis on unique song structures. Expect to hear a lot of odd time signatures and instruments uncommon to metal in this genre as well as its many crossovers with other genres. Prog metal is exemplified by a wide range bands, notably Opeth, Dream Theater, and Fates Warning.
- Power metal is a point of confusion for plenty of metal listeners, as it's used to describe both classic heavy metal, usually of the heavier variety hailing from the US or Germany, as well as a very different modern offshoot. The modern offshoot(dubbed "euro power metal") is nowadays closely related to progressive metal and features bands such as Rhapsody, Symphony X, and Angra. This brand of theatrical, keyboard-laden, sometimes even orchestrated metal was pioneered by groups such as Edguy and Helloween.
- Metalcore is advertised as a combination of heavy metal and hardcore punk(hence "metalcore"), but this can be misleading. Older "metalcore" bands were essentially hardcore punk, but had more metallic - usually meaning more technical - riffing. Look at Integrity and Earth Crisis for examples. Metalcore bands nowadays are indeed influenced by the metalcore bands of old(perhaps not Integrity, but certainly bands such as Hatebreed or Agnostic Front), but mainly in the way they use breakdowns. Metalcore songs usually feature segments where they transition to slower, "chugging" riffs. These are the breakdowns, and they're meant to make the song seem heavier and more intense as a whole. My impression of most metalcore bands, however, is that outside of the breakdowns, the primary influence is Gothenburg-styled melodic death metal, which is itself usually inspired by traditional heavy metal and thrash metal.
- Just to be clear, Nu metal is a touchy subject. Like metalcore and hair metal, it's something of a pariah here. It is not well liked on 235 or, indeed, in underground metal communities in general. Mainstream publications such as All Music Guide and even Wikipedia tend to classify it as a kind of metal, and understandably so, but metal communities rarely acknowledge it as one of metal's many subgenres. The reasons for this are complex. Stronger arguments involve the actual similarity nu metal has to the "canon" genres of metal. Nu metal tends to be aggressive, and so does metal. Nu metal tends to feature distorted guitars which the songs are often written around, and so does metal. Nu metal tends to have loud, double bass-heavy drumming, and so does metal. In fact, it is reasonable to say that nu metal was influenced, perhaps quite dramatically, by the "groove metal" style which emerged in the early 90s, which is acknowledged, sometimes grudgingly, as truly metal. Look at bands such as Pantera and Testament in their Low era for examples. Though a little late for the craze at the height of its popularity, the nu metal band Damageplan actually formed from the ashes of Pantera. I think this is probably the strongest argument nu metal supporters can make in favor of it being "really" metal.
Me, I like hard rock/metal depending on the Bands but I always favor all metal, so its hard to narrow it down what type of metal I prefer but to be perfectly honest, Heavy Metal is my go to and so is Nu Metal. Heavy Metal gets my vote.