It's unfortunate that the term 'Skinhead' has become so closely linked with far-right that the two terms are almost interchangeable, an example of which is here in a popular British newspaper.
The original Skinheads grew from West Indian Rude Boy and British Mod fashion, music and culture in the '60's. Black and white kids mixed and race wasn't a part of the scene. This continued through the decline of the Skinhead culture in the early '70's. It wasn't until Skinhead styles were adopted and adapted by some Punk's that there was a kind of Skinhead revival in the late 70's, and it became associated with the Far-Right / Neo-Nazism and football hooliganism. Do a Google Image Search for Skinhead and it's pretty ugly.
It's a great shame that association is still around, as Skinhead style is amazingly sharp, mixing several fashions with working-class influences to produce something that's instantly recognisable.