"'눈치 9단'이라는 말 무슨 뜻입니까?"
눈치 (눈: eye, 치: subtle expression, look, etc) can mean two related things.
1) a quick sense to assess a situation and do the right thing, especially to quickly read what people want and do it to please them.
- 눈치가 빠르다 - be quick witted in adapting to a situation.
2) subtle awkward or inhospitable expressions and actions people display toward an unwelcome/uncomfortable person or a group of people.
- 눈치를 받다 - be subjected to subtle inhospitality.
눈치 9단 means that someone's 눈치 (in the sense of #1) is at 9단 (9th grade/level), which is the highest one can go (9th is the highest level in the board game 바둑 - "go").
So it means the person has a highly developed sense of adaptability.