to apologize — to express regret or say sorry for a mistake or offense.
遅刻したので彼に謝った。
I apologized to him because I was late.
まずはきちんと謝ることが大切だ。
First, properly apologizing is important.
彼女は誤解を招いたことについて深く謝った。
She apologized deeply for causing the misunderstanding.
(dated) to refuse or decline — an old usage where 謝る can mean to refuse an offer.
古い手紙では『その申し出を謝る』と書かれていた（＝断る）。
In an old letter it was written 'to謝る the offer' (i.e., to refuse).
(dated) to be unable to bear; to be defeated or at a loss — an archaic meaning now rarely used.
古文で『敵に謝る』とあるが、現代語では通じにくい（＝敗れるの意）。
Classical texts say 'to 謝る to the enemy,' which is hard to understand in modern Japanese (meaning 'to be defeated').