Learn Russian: Почему вы думаете, что договора нет? - Why do you think that there is no contract?

This is the Genitive case! When you use нет to say that something doesn't exist or isn't there, the thing that is missing must be in the Genitive case. The dictionary form of the word is договор (a masculine noun ending in a consonant), so we add -а to make it договора.

Why is нет at the end of the sentence? Shouldn't it be нет договора?

In Russian, word order is very flexible. You can absolutely say нет договора, but putting нет at the very end (договора нет) is a common way to emphasize the fact that the contract does not exist.

Wait, doesn't что mean "what"? Why is it translated as "that"?

что (always pronounced што) is a very useful word! It means "what" when asking a question, but it also acts as a connecting word meaning "that". We use it here to link the first part of the sentence to the second part.

Why does the verb end in "-ете" (думаете)?

Because the speaker is using the pronoun вы (you). In Russian, вы is used either when talking to multiple people, or when speaking politely and formally to one person. The dictionary form of the verb is думать (to think), and it changes its ending to -ете to match вы.