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What question is asked for response muy bien?

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"Muy bien" just means "very well". So, any question that calls for this answer would work, like:
(1) How are you? -- ¿Cómo estás?
(2) Does this work? -- ¿Funciona esto?
(3) Do you know this area? -- ¿Conoces esta area?
(4) How does your sister sing? -- ¿Cómo canta tu hermana?
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How are you today?
or
How does the soup taste?
or
How did you do on the test?
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Esta muy bien?

As written, it means "this very good". Probably you mean "Está muy bien", meaning "you are very good" or "it is very good", or "he/she is very good". Context woud determine e

What is mui bien?

It's spelled wrong. It should be muy bién and it means very good.