Translation:One glass, please.
A コップ (koppu) is the general term to refer to a container/vessel used for drinks.
A カップ (kappu) is a コップ (kappu) with a handle used specifically for warm/hot drinks.
A グラス (gurasu) is a コップ (koppu) made from ガラス (garasu) referring to a cup without a handle used only for cold drinks.
No, it wouldn't be "Can I get one cup please", because that is incorrect. "Kudasai" literally means "Please give me", not "Can I get". If I am asking for something, I'm not going to say "Can I get" unless I want to go get it myself. If I'm in a restaurant and I want some water, I'm not going to say "Can I get some water" unless I want the server to point to the water dispenser. "Can I get" is an incorrect translation, and "Please give me" does not sound demanding at all. Besides, it is the accurate translation of "kudasai".
コップ = glass (drinking vessel); tumbler, drinking glass, From Portuguese “copo”, From Dutch; Flemish “Kop” cap isn't totaly wrong but the japanese word for cap is "カップ ". I f you want to hear "cap" please use this word. There are several ways to translate ~te kudasai. And I think "Can I get ...(please), is not the best one (my opinion). If you want to hear only that phrase, you should say so.
From Dutch kop and also from Portuguese copo. Cognate (via Late Latin cuppa) with カップ (kappu), from English cup.
The phrase "コップを一つください" is a polite way to say "Please give me a cup." The English question "Can I get one cup?" Is unnatural to me, and indicates I want to get up retrieve the cup myself. "Kudasai" literally means "Please give me", and not "Can I get". This is prevalent throughout the Japanese course (which is otherwise excellent).
So, how do we get someone to fix this?
Okay so glass, is コップ. Then what is ガラス?！ You can't just say it's only the glass for windows and stuffs not the proper glass we drink water in it, because I have actually heard people saying コップ to cups, and ガラス to glasses. Can someone just explain this to me?! I really don't get it... Maybe I'm wrong?
～杯 fits better for glasses。一杯のコップ【いっぱいのコップ】