Learn Russian: У вас есть бумага? Нет, бумаги нет. - Do you have paper? No, there is no paper.

This happens because of the second нет (meaning "there is no"). In Russian, when you state that something does not exist or is absent using нет, the missing item must be in the Genitive case. For a feminine noun ending in -а like бумага, the Genitive ending is -и, which gives us бумаги.

Why are there two нетs right next to each other?

They have two different jobs! The first нет simply means "No" (answering the yes/no question). The second нет is the verb-like word meaning "there is not" or "is not present". A literal translation would be "No, paper is absent."

If the English translation is "Do you have paper?", why isn't the Russian word for "you" the subject?

Russian doesn't use a verb for "to have". Instead, it uses the structure У вас есть..., which literally means "At you there is...". Because "paper" is the thing that "is" there, бумага is the actual subject of the first sentence. That is why it stays in its basic dictionary form (Nominative case) in the question.

What case is вас in the first sentence?