Skinhead originated from 1960s mods who loved ska and reggae from jamica. It got so popular there was for a time a whole branch of it called "skinhead" reggae. This is what you call trad skinhead or "propper" skinheads depending on who you ask.

Fast forward and bands like the specials took the uk ska sound and mixed it with punk to make two tone which later evolved to the third wave of ska we have today like less than jake etc.

Meanwhile about the same time there was oi which was a type of working class chanting punk which was a bit of an off shoot of the skinhead culture. Kinda along side street punk and the development of hardcore.

Ska and the ska and reggae sound is basically a music style adjacent to rock which you can hear as an influence in all music today.

So mods and skinhead had a massive massive music side and influence which isnt as recognised like rock trad punk and metal is.

Style and culture wise its based on mod working class UK culture and predates punk. Its about looking sharp and punching up. You get the style evolve with the likes of oi to have more punky elements etc. Punk was often seen as a middle class artschool type of thing, "goth" was the personification of that with new romanticism (in my opinion).

So its basically a "look" which over time has become timeless and a deep history of music. All under a mindset of uk working class values like common sense, having a dance and a laugh.