Learn German: Mach den Mund auf!

In English, we use possessive words with body parts like "your mouth". In German, it is much more common to use the definite article (like den here) when it is obvious whose body part is involved. Since you are giving a command to someone directly, it is understood they should open their own mouth!

Where does the word auf at the end come from?

It is the prefix of the separable verb aufmachen (to open). In commands like this one, the prefix splits off from the main verb (mach) and goes to the very end of the sentence.

Why is the verb just Mach without a normal ending like "-st"?

This sentence is a command, also known as the imperative. To tell one person you know well (a du relationship) to do something, you take the normal du form of the verb (du machst) and drop both the word du and the "-st" ending. This leaves you with just Mach.

Why is it den Mund and not der Mund?

The word for mouth is a masculine noun (der Mund). Because the mouth is the direct object of the sentence (the thing that is being opened), it must be in the accusative case. In the accusative case, the masculine article der always changes to den.