"This computer is not necessary."
Translation:Ten komputer nie jest potrzebny.
why is the nominative case used after jest when it's an adjective, but when it's a noun you use the instrumental case?
Adjective is inflected to the same case as followed noun. There you have "ten komputer" - nominative, so adjective has to be also in nominative (potrzebny/niepotrzebny). In case of sentence: "Nie jest potrzebnym komputerem" you have noun and adjective in instrumental.
but when it's something like "mężczyzna jest człowiekiem", one is in nominative case and the other 1 is in instrumental
Mężczyzna - nominative, main case, because you are describing him
człowiekiem - instrumenal, used if you want to say "is who?", "is what?" (and also if you are doing something with it)
mężczyzna jest człowiekiem - instrumental
mężczyzna jest silny - nominative
why are they in different cases?
Because "człowiekiem" is a noun, so its inflection isn't connected with "mężczyzna". Adjectives are connected with nouns and they have to have the same case ("silny mężczyzna" - "mężczyzna jest silny", or "jest silnym mężczyzną - both adjective and noun in instrumental case)