There really is no definitive way to correctly categorize bands into empirical genres. However, there are various ways that people classify various musical forms.
Classical music is not something that I am particularly familiar with, but it generally uses traditional acoustic instruments in either an orchestral or chamber presentation.
Jazz is a term used for a form of music that evolved from Ragtime, Classical Music and Blues. It is ethnically tied to Black Americans, but there are other fusions of Jazz that have developed. There are several sub categories in Jazz, including Fusion, Swing, Bebop, Hard Bop, Post Bop, Avante Garde, Smooth, Contemporary, and Traditional to name a few.
Blues is a music that has a characteristic call/response format. However, while there are standard forms of blues, such as the I-IV-V progression, it has many different forms and styles that are as unique as the person playing it. For example, Blind Willie Johnson and Robert Johnson are both delta bluesmen, but Robert Johnsons style of playing is very much different than Blind Willie.
Folk Music is a broad genre that consists mainly of traditional songs that are usually played on acoustic instruments in either a solo or ensemble setting. It has many ties to blues and country/bluegrass (at least in America). Every country has it's own folk songs as well, such as "Whiskey in the Jar" (a traditional Irish folk song).
Rock developed in the 1950's as an offshoot of Blues/Rhythm and Blues. It uses similar progressions and scales as blues, but is characterized to some as an up tempo version thereof. It developed the use of heavily distorted guitar sounds and long solos.
Metal developed from Blues and Rock music, but has a different aesthetic to it. Black Sabbath is to many, the first metal band, characterized by heavy, downtuned guitars/basses, and misanthropic, violent, and angry lyrics. There are many offshoots and subgenres of metal that have developed, but the heavily distorted, downtuned guitars and lyrical content is a common characteristic of most metal.
Punk developed in the late 60's and early 70's from early rock and roll and garage rock, like The Kinks, The Stooges and The MC5. It was characterized by the DIY mentality and a general 'lack of skill' among practitioners. Very seldom did songs have solos. Generally, they consisted of barre/powerchord styles, as well as heavily distorted guitars. It was generally a reaction to the political and social atmosphere of the places where it developed. Punk music generally has angry, disillusioned/rebellious lyrics.
I gave a definition of Hip Hop in the other thread.
Pop music is shorthand for 'Popular Music'. What ever happens to be popular at the time is pop music. However, modern pop music is generally characterized by simple melodies and 'creative' 'emotional' lyrics.
I hope that helped a bit.
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Listen here Livin just to keep from dyin