"Do you know about something" or "do you know how to do something" has already been covered with ㄴ/는지 알아? but this one is slightly different. What makes it different is instead of asking if someone knows about or what something is, this is more focused on how to do something. Think of abilities (능력) and ways of doing things (방법).
Examples:
젓가락을 사용할 줄 알지요?
You know how to use chopsticks, right?
영어 할 줄 아세요?
Do you know how to speak English?*
아이폰을 쓸 줄 몰라요
I don't know how to use an iPhone.
그는 불안해서 어쩔 줄 몰라 했어요.
He was so embarrassed that he didn't know what to do.
너무 좋아서 어쩔 줄 모르겠어요
I'm so happy I don't what to do.
A: 일본어 할 줄 몰라? 거기서 오래 살았잖아.
You don't know how to speak Japanese? You lived over there for a long time, you know...
B: 그래. 일본어 못 해. 한국어는 할 줄 알아?
Yeah, I can't speak Japanese. So what? Can you speak Korean?
*compare this with 영어를 할 수 있으세요? Can you speak English? Both ask the listener if they feel comfortable speaking English as a foreign language but each has its own feeling. 할 수 있다 might be a little more formal sounding but both virtually mean the same thing despite the English translation's hair-splitting difference.
