To work for or be employed at (a company/organization). (Usually written 勤める.)
彼は東京の銀行に勤めている。
He works at a bank in Tokyo.
大学を卒業して市役所に勤めた。
After graduating from university, I worked at the city hall.
長年同じ会社に勤めるつもりだ。
I intend to work for the same company for many years.
To serve as or act in a role; to fill a position (e.g., serve as host or chairman). (Usually written 務める.)
来賓の紹介を私が務めます。
I'll be responsible for introducing the guests.
彼女はイベントの司会を務めた。
She served as the host of the event.
校長として二十年務めた。
He served as principal for twenty years.
To conduct a religious service or perform ritual duties.
神父がミサを勤めた。
The priest conducted the Mass.
毎朝お寺で勤める時間がある。
There is a time in the morning to perform the rites at the temple.