Learn Icelandic: Sýn mér bílinn!

Sýn comes from the verb sýna (to show). To give a command to one person, you usually just drop the -a at the end of the infinitive verb. So, sýna becomes the command sýn.

I often see commands ending in "-du" or "-tu". Why doesn't this sentence say sýndu?

In spoken Icelandic, it is very common to attach the word for "you" (þú) to the end of the command, which would make it sýndu. Using just the base command sýn is perfectly correct grammar, but it sounds a bit more direct or formal than sýndu.

Why do we use mér instead of the accusative mig for "me"?

The verb sýna requires the dative case for the person receiving the action (the person you are showing something to). Because of this rule, we must use the dative form mér instead of mig.

What case is bílinn in this sentence?

It is in the accusative case. When using the verb sýna, the thing being shown is the direct object and takes the accusative case. Since bíll (car) is a masculine word, the definite version (the car) drops an "l" and changes from the dictionary form bíllinn to bílinn.