"This skirt is not new."
It is actually part of the adjective. Simply put, for い-adjectives, く forms the base for negative conjugations. E.g. あたらしい (新しい) "it's new" -> 新しくない "it isn't new".
Notice how the negative form itself becomes a new い-adjective, which in turn can be conjugated for past tense: 新しくなかった ("it wasn't new")
It turns the adjective into an adverb (like ~"ly" does in English), so it's technically 新しい as an adjective modifying the verb ない. If you translated 新しくない literally, it would be something along the lines of "it new(ly) isn't" or something like that (Though that's obviously grammaticaly incorrect in English, lol)... It's easier to think of "新しくない" as the negative form of 新しい and nothing else, of course, but this is how it actually works.