Learn German: Mach das Buch auf!

This sentence uses the separable verb aufmachen (to open). In German, when you use a separable verb in a normal sentence or a command, the core part of the verb (mach) takes the normal verb position, and the prefix (auf) gets kicked to the very end of the sentence.

I thought machen meant 'to make' or 'to do'. Why does this sentence mean 'open'?

You are exactly right about machen! However, when you add the prefix auf to create the separable verb aufmachen, the meaning changes to 'to open.' You can think of it literally as 'making something open.'

Why is the verb just Mach without an ending or a subject?

This sentence is a command (imperative) directed at someone you would address as du (informal 'you'). To give a command to a du, you take the normal verb form (machst), drop the -st ending, and completely leave out the word du.

Is das Buch in the accusative case here?

Yes, it is the direct object receiving the action of being opened, so it is in the accusative case. Because Buch is a neuter noun, its 'the' word stays as das and doesn't change shape like masculine nouns do.