Learn German: Verstehen Sie den Vertrag der Firma? - Do you understand the company's contract?

This is the Genitive case in action, which shows possession. For feminine nouns, the article die changes to der in the genitive. So der Firma translates to "of the company" or "the company's".

Shouldn't there be an -s at the end of Firma for the genitive case?

No, only masculine and neuter nouns get an -s or -es ending in the genitive. Feminine nouns like die Firma do not get an extra ending, so the article does all the work to show the case.

Why is the first article den instead of der?

Vertrag (contract) is a masculine noun. In this sentence, it is the direct object—the thing being understood. That means it must be in the accusative case, where the masculine article der changes to den.

Why does der Firma come after den Vertrag? In English we put "the company's" first.

In German, the genitive phrase almost always comes directly after the noun it belongs to. You are essentially phrasing it as "the contract of the company" (den Vertrag der Firma).