In spoken Korean, 뭐 is used instead of 무엇.
무엇 ⇒ 무어 ⇒ 뭐
It's a demonstrative pronoun.
저거 뭐예요? What's that?
거기서 뭐 먹었어? What did you eat there?
어제 뭐 했어? What did you do yetsterday?
뭐 잘못된 거 있어? Is there anything wrong?
무슨 is prenominal, you can use it when you're asking something you don't know what it is.
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?
You can use 어떤(a shortcut for 어떠한, 어떠하다 = "to be like what") when you ask about something or someone, especially characteristics, personality, conditions, contents/substance. It's also used as "some" as in "some people".
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?")
You can use 어느 when you're asking which one it is.
어느 거? / 어느 거 말이야? Which one? (among those things)
어느 나라 사람이에요? Where are you from?
(His/her country is supposed to be one of all the countries, 어떤 나라 사람이에요? doesn't make sense unlike 어떤 나라예요? does.)