To bow or lower one's head as a gesture of greeting or respect (covers slight nods to deep bows).
来客が来たら皆で頭を下げて迎えた。
When the guest arrived, everyone bowed to welcome them.
会議の最初に社長が頭を下げて挨拶した。
At the start of the meeting the president bowed and greeted everyone.
店員が深く頭を下げてお詫びした。
The clerk bowed deeply to apologize.
To apologize; to show remorse by lowering one's head (an act of apology).
彼は自分のミスを認めて、頭を下げた。
He admitted his mistake and apologized (bowed).
トラブルについて直接会って頭を下げるつもりだ。
I intend to meet them in person and apologize for the trouble.
公共の場で謝る時は形式として頭を下げることが多い。
When apologizing in public it's common to bow as a formality.
To yield to someone, to show deference, or to admire someone (figuratively 'to bow to' someone).
若手選手の才能にみんなが頭を下げた。
Everyone bowed to (admired) the young player's talent.
彼女の知識の深さには頭を下げざるを得ない。
One can't help but respect the depth of her knowledge.
経験豊富な先輩には自然と頭を下げてしまう。
You naturally defer to a senior with lots of experience.