Which case is this and why?
This is genitive, as „potrzebować” belongs to a group of verbs that goes with genitive. Somebody else might give you more detailed info.
So that verb is governed by the Genitive case, then?
Yes, I believe this is what I basically wrote above.
Shouldn't such a sentence need imparative (is this the name in English as well? I mean the case you use to order people.)?
But imperative in English very rarely mentions the subject, so it would almost certainly be "Need coffee!".
Moreover, that's a pretty unusual thing to say.
Although technicly possible almost no one uses imperative forms of "potrzebować" like: potrzebuj or potrzebujcie. That's one thing.
This sentence doesn't make use of imperative. This is rather an ovservation/advice. (You look exhausted). You need coffee!
How do I know it is in the plural? I answered it correctly in the singular.
If Duolingo asks you to translate an English sentence with "you" into Polish, you are allowed to pick either singular or plural. In real life you would need proper context, though.
Objects of the verb potrzebować take the genitive case, not the accusative. The accusative is kawę, and the genitive is kawy.
Please read the comment section. This has already been discussed.