1To work for or be employed at a company, government office, school, etc. Typically used in the pattern 〜に勤める to indicate one's workplace, and usually written 勤める. Implies a continuing, formal employment relationship — slightly more formal than 働く.
父は東京の銀行に勤めています。
My father works at a bank in Tokyo.
大学を卒業してから、ずっと同じ会社に勤めている。
I've been working at the same company ever since I graduated from university.
どんな会社に勤めたいですか。
What kind of company do you want to work for?
母は近所の小学校に長年勤めてきた。
My mother has worked at the nearby elementary school for many years.
2To serve as, act as, or fill a specific role or position — host, chairperson, lead actor, etc. Usually written 務める and used in the fixed pattern 〜を務める.
彼女が今日のイベントの司会を務めます。
She will serve as the host of today's event.
次回の会議の議長は田中さんが務めることになった。
Mr. Tanaka has been chosen to chair the next meeting.
彼はその映画で主役を務めた。
He played the lead role in that movie.
彼は十年間、社長を務めた。
He served as president of the company for ten years.
3To conduct or perform a religious service — Buddhist sutra recitation, memorial rites, a Catholic Mass, etc. Written 勤める. Used in religious or temple contexts rather than everyday speech.
住職が祖父の一周忌の法要を勤めてくださった。
The head priest conducted the first-anniversary memorial service for my grandfather.
神父がクリスマスのミサを勤めた。
The priest conducted the Christmas Mass.