To shut one's eyes; to close one's eyes (physically).
夜、寝る前に目をつぶって深呼吸をした。
At night, I closed my eyes and took a deep breath before sleeping.
眩しくて思わず目をつぶった。
It was so bright that I instinctively shut my eyes.
写真を撮る時に目をつぶらないでください。
Please don't close your eyes when the photo is taken.
To turn a blind eye to something; to deliberately overlook or pretend not to notice (a fault, wrongdoing, etc.).
上司は部下の経費の不正に目をつぶった。
The boss turned a blind eye to the employee's expense fraud.
学校はこの問題に目をつぶらず、きちんと対応するべきだ。
The school should not ignore this issue and should deal with it properly.
小さな違反に目をつぶると、あとで大きな問題になることがある。
If you overlook small violations, they can become big problems later.
To die (euphemistic), often used as a gentle way to describe someone's passing (closing one's eyes).
祖父は静かに目をつぶった。
My grandfather quietly closed his eyes (passed away).
彼女は老衰で穏やかに目をつぶったと聞いた。
I heard she peacefully closed her eyes from old age.