Можно is pretty much the same thing as the English verb 'can'.
Можно мне кофе? Can I have a coffee?
...
Of course there are many ways to order things in Russian; it's just about the same as in English.
You can just bark "чёрный кофе, и побыстрее, я тороплюсь" (which is 'coffee, and quickly, I'm in a hurry') to a waiter and the only his responce expected is saying "Ok" and running for your coffee. Though things like that are rather rude.

You can even say just "кофе" and that's just fine, but oops, that's where you're at a risk of the waiter falling into a wordy explanation of what sorts of coffee they can offer to you.