1. What does it mean?
눈치 is an established Korean word for sensing what people feel or what a situation calls for without everything being stated. 눈치가 빠르다 describes someone who picks up such cues quickly.
Also written or said as nunchi, nun-chi
2. The nuance
It is not a magical ability or a trait every Korean person shares. 눈치를 보다 can describe worrying about other people's reactions, so translating every use as the positive 'social intelligence' misses part of the meaning.
3. Examples
“친구가 말하기 싫어하는 눈치라서 다른 이야기로 넘어갔어.”
My friend seemed reluctant to talk about it, so I changed the subject.
“작은 질문까지 눈치 보지 말고 편하게 물어봐.”
Don't worry about how every little question will come across—just ask.
Sources
JOINUS KOREA: Korean language learning
References support the meaning and usage. Example sentences are written for NoCapWiki.