The face or visage — the front part of the head (eyes, nose, mouth), i.e., someone's physical facial appearance.
彼女は小さな顔をしていて、写真映えする。
She has a small face and photographs well.
鏡で顔を確認してから出かけたほうがいいよ。
You should check your face in the mirror before going out.
寒さで顔が赤くなった。
My face turned red from the cold.
その人は顔立ちがはっきりしている。
That person has distinct facial features.
One's look or expression — the countenance that shows emotions (e.g., a happy face, a troubled expression).
彼は嬉しそうな顔をしていた。
He had a happy expression on his face.
先生は生徒の反応を見て深刻な顔をした。
The teacher looked serious when seeing the students' reactions.
その知らせを聞いて顔が輝いた。
Her face lit up upon hearing the news.
彼女は何かを隠しているような怪しい顔をしていた。
She had a suspicious look, as if she was hiding something.
Honor or face (social reputation); used in idioms like 'to save face' or 'to lose face'.
上司の顔を立てて、彼の案に賛成した。
I supported his proposal to save face for my boss.
彼の失敗で会社の顔がつぶれたと感じた。
I felt the company's reputation was damaged by his failure.
大事な取引先の前で彼の顔をつぶすわけにはいかない。
I can't embarrass him in front of an important client.
Influence, clout, or reputation — the advantage or authority someone has because of connections or status ('to have influence', 'to be well-connected').
彼は業界で顔が利くので、すぐに手配してくれた。
He has clout in the industry, so he arranged it quickly.
顔が広い友人のおかげで会場を借りられた。
Thanks to a well-connected friend, we were able to rent the venue.
この仕事は彼の顔で受注したと言ってもいい。
You could say we won this contract because of his connections.