A person's mood, temper, or spirits—how they are feeling emotionally at a given time.
今日は部長の機嫌が悪くて、会議が早く終わった。
The manager was in a bad mood today, so the meeting ended early.
子供の機嫌を取るためにお菓子を買った。
I bought sweets to cheer up the child.
彼は機嫌によって態度が変わることがある。
His attitude sometimes changes depending on his mood.
(Somewhat older or broader usage) a person's condition, health, or general state—used to refer to how someone is doing physically or situationally.
祖母の機嫌が優れないので、念のため病院へ連れて行った。
My grandmother wasn't feeling well, so I took her to the hospital just in case.
高齢者の機嫌や体調を常に確認することが大切だ。
It's important to always check the elderly person's condition and state.
最近は機嫌がいいようで、よく食事をする。
Lately they seem to be in good condition and eat well.
Used (often as ご機嫌) to mean being in a good mood or high spirits; a polite/exclamatory way to ask or remark about someone's state.
祖母はご機嫌で昔話をたくさん話してくれた。
Grandma was in high spirits and told us many old stories.
ご機嫌いかがですか？ （目上の人や客への挨拶）
How are you? (a polite greeting to a superior or guest)
彼女はご機嫌な顔で歌を歌っている。
She is singing with a cheerful expression.