"Le serveur me donne le café" would be "...the coffee".
"du", "de la", "des", "de l' " are often literally translated as "of the" or "from the". But when they are paired with a countable noun (meaning anything that you can have "some of" or "all of") they become partative articles, and they mean "some but not all" or just "some".
So this sentence can be translated to "The waiter gives me some coffee". Or, "The waiter gives me some of the coffee." In English we can drop the "some" and still arrive at the same meaning; so, "The waiter gives me coffee."
Because of that difference, you'll see a lot of this type of translation where you'll need to add a "de" into the sentence in French:
I have apples. --> J'ai des pommes. He threw sand in her face!--> Il a jeté du sable dans son visage! There are cats in the house.--> Il y a des chats dans la maison.