Learn Swedish: Nej, vi har ingen ost kvar. - No, we have no cheese left.

Ingen is the form of "no" or "none" used with en-words (nouns that use en). Since the word for cheese is en ost, we use ingen. If the noun were an ett-word like vatten (water), you would use inget instead (inget vatten).

Could I say vi har inte ost kvar instead?

Swedes strongly prefer using ingen, inget, or inga with nouns to express the absence of something. Saying vi har ingen ost (we have no cheese) sounds much more natural and idiomatic than trying to say vi har inte ost (we do not have cheese).

Can ingen also mean "nobody"?

Yes! When ingen stands alone as a pronoun without a noun following it, it means "nobody" or "no one". But in this sentence, it is placed directly before a noun (ost), so it acts as a determiner meaning "no" (as in "no cheese").

How does the word kvar work here?

Kvar means "left" or "remaining". It is placed after the object it describes, acting exactly like the English word "left" does in the phrase "no cheese left".