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What

뭐 (adverb, demonstrative pronoun)

지난 주말에 뭐 했어요? – What did you do last weekend?
내일 뭐 하고 싶어요? – What do you want to do tomorrow?
이것은 뭐야? – What is this?
이름은 뭐에요? – What is your name?
저거 뭐예요? – What’s that? 
거기서 뭐 먹었어? – What did you eat there? 
어제 뭐 했어? – What did you do yesterday? 
뭐 잘못된 거 있어? – Is there anything wrong? 
뭐 부를래? – What do you want to sing? 

무엇 (noun)

점심을 먹었어? 무엇을 먹었어? – Did you eat lunch? What did you eat?

무슨 (adjective, prenominal (which))

무슨 말이에요? – What do you mean? 
무슨 뜻이에요? – What do you mean by that? 
무슨 일이에요? – What’s the matter? 
무슨 책이에요? – What kind of book is it? 
무슨/어떤 자동차 입니까 ? – Which car is it ?
무슨 노래 부를래? – What song do you want to sing? 
무슨 일을 하고 싶어요? – What work do you want to do? (what job do you want to have?)
무슨 영화를 보고 싶어요? – What movie do you want to see?

무슨 종류의 – Which kind of, sort, type
무슨 종류의 음식 좋아하세요 ? – What kind of food do you like?

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Which 

어떤 (어떠한, 어떠하다) – what, which, to be how, to be like what
also “some” (some people)

characteristics, personality, conditions, contents/substance of something or someone. 

그 사람 어떤 사람이에요? – What is s/he like? 
어떤 사람이 되고 싶어?-  What kind of person do you want to be? 
어떤 게 나을까? – Which one would be better? 
어떤 사람들은 그 노래 안 좋아해 – Some people dislike the song. 
넌 어떤데? – What about you? (not as in “I’m fine, what about you?”) 
어제 어떤 외국인들을 봤어요 – Yesterday, I saw some foreigners
어떤 영화를 보고 싶어요? – which movie do you want to see? 

어느

for place/location you must use 어느 instead of 어떤:

어느 거? / 어느 거 말이야? – Which one? (among those things) 
어느 나라 사람이에요? – Where are you from? 
어느 노래 부를래? – Which (of these) songs would you like to sing? 
어느 집에서 살아요? – Which house do you live in?
어느 대학교를 다녀요? – Which university do you go to?