Learn Danish: En øl, tak! - A beer, please!

Danish doesn't have a single direct translation for the English word "please". When you are asking for something, especially when ordering food or drinks, you just add tak at the end of the request. It literally means "thanks", but it functions exactly like "please" in this situation.

I thought øl was an uncountable noun. Why does it use the article en here?

You are right that the liquid itself is uncountable, and when talking about beer in general, it is a neuter word (taking et). However, when you are ordering a specific serving of beer, like a glass or a bottle, it becomes a common-gender noun (en øl). You will hear en øl very often in bars and restaurants!

Could I say Jeg vil gerne have en øl instead?

Yes, absolutely! Jeg vil gerne have en øl means "I would like to have a beer" and is a very polite and common way to order. Saying En øl, tak! is just a shorter, casual, but still perfectly polite way to ask for a drink.