Making a complete list of all metal styles and genres today is nearly impossible as there are not only the big metal genres like death, power and trash around but also hundreds of variations and mixtures of those. The same reason and the fact that many bands play music from several styles or change it time makes it imposible to clearly define the style of a band. Thats why its mostly up to the listener to choose in wich category a band falls.
With Heavy Metal it all begun... thats why its often called "True Metal" or "Classic/Traditional Metal". Most metal bands of later genres (that didn't come up before the mid eighties) are at least partially inspired by the big and famous "metal pioneers" like Iron Maiden or Judas Priest. The music focuses on aggressive riffs, two-guitar harmonies and powerful clean vocals with emphasis on the upper ranges and the songs often deal with Metal as a way of life.
A side movement of the Heavy Metal is the New Wave Of British Heavy Metal (NWOBHM) that rose in the early 80s in England. For example Iron Maiden and Judas Pries belong to this style and its in general a "rockier" variation of the early metal sounds.
Examples: Iron Maiden, Black Sabbath, Judas Priest, Manowar
Trash metal came into existence in the late 80s and was at this time very popular. It relied on a fast, choppy riffing style and what was in those days the fastest machine-gun drumwork and the vocals were usually some sort of shout. Only some bands like Forbidden or Agent Steel used melodic singers. With the rise of Death Metal in the late 80's Trash declined in popularity.
Examples: Metallica, Slayer, Megadeth, Anthrax, Sodom
Power Metal, in its usual form, consists of fast songs with an emphasis on melody, often accentuated with keyboards. Vocals are melodic and usually high-pitched, and the use of the double-bass drum kick is practically mandatory. As a style, Power Metal tends to use the guitar as a melodic, rather than a rhythm instrument, though there are exceptions. Lyrical themes are largely fantasy based and are often accused of being too ‘Happy'. Power Metal comes nearly entirely from Europe and there are even regional variations of the basic Power Metal style: German Power Metal is mostly heavier and uses less keyboards, Italian Power Metal is melodic with high-pitched vocals and often very fast and Swedish Power Metal is considered to be a mix between those two.
Many bands also mixed Power Metal with other genres like trash, death or doom and produced new styles like the very fast Speed Metal or the Symphonic Metal.
Examples: Helloween, Gamma Ray, Blind Guardian, Rhapsody, Dragonland
Progressive Metal is a mixure of the progressive rock of the 70's (i.e. Rush and King Crimson) with the sound of Heavy Metal. The product were melodic but more musically challenging and intricate sounds. The founders were Queensryche and Fates Warning in the mid 80's, but the most influental band in the genre was Dream Theater. They added keyboard layers and very complex, over the top songwriting to the Fates Warning approach.
Examples: Queensryche, Fates Warning, Dream Theater
Death Metal emerged from Trash between 1985 and 1987, after the first Death Metal releases by Slayer, Death and Possesssed, and boomed afterwards, especially in the United States. Swedish Death took the fast, technical assault of US Death Metal and added a slash of melody and a slightly slower, heavier overall sound. This was largely the legacy of the legendary Sunlight Studios where Entombed produced their buzzsaw guitar sound that Death Metal bands are still imitating today.
Examples: Possessed, Slayer, Death, In Flames
Early on two basic schools of Black Metal emerged: the "Raw" style, which imitates the primitive instrumentation and raw production of early Black Metal. (Exemplars would be Dark Throne and Dark Funeral, plus legions of others.) And the "Melodic" school (sometimes called "Symphonic") which explored the use of keyboards and more melodic songwriting to create atmosphere. (Adherents would include Emperor and Dimmu Borgir.)
Black Metal exploded in popularity in the mid to late 90's, and while the tide has somewhat receded, the genre is still hugely popular. While Norway was the epicenter of the movement, now Black Metal scenes exist all over Europe, as well as healthy scenes in Japan, South America, and even Australia.
From the beginning, when Norwegian Black Metal musicians made international news by burning churches and murdering each other, the genre has been known for extreme beliefs, and in fact Black Metal is the only genre with a definite ethos as part of its image. True, Death is usually about violence and/or evil, and Doom is usually about something depressing, but Black Metal is expressly about evil and hatred, never anything else.
Examples: Dark Throne, Satyricon, Burzum, Emperor, Enslaved