"The apple is not black."
Translation:Az alma nem fekete.
No; you can't use van here.
In English, a predicate (the bit of a sentence that talks about the subject and what it is or does) is essentially always a verb, but in Hungarian, a noun or an adjective can be a predicate by themselves.
In Az alma fekete, the adjective fekete is the predicate. So in that sentence, you would translate it not as merely "black" but as "is black".
Similarly in Duo's sentence: nem fekete turns into "is not black" when you translate it into English.
Hopefully this will help some: http://www.hungarianreference.com/Van-is-exists-omitting.aspx
I'm confused about the word order. In previous examples it looked like it would have been "nem fekete az alma"?