To add to it, 요청, 요구, 수용하다, 수락하다 are formal, hanja-based words.
받다(receive), 받아들이다 (accommodate) are native, everyday words.
받다 is different from 받아들이다 in that it means "receiving" a request, not accepting it.
Another common word is 들어주다, which means "accommodate and act on a request".
받아들이다, 수용/수락하다 is "agreeing" to do it, while 들어주다 implies actually doing it.
The opposite word is 거절하다 (decline), and less commonly 거부하다 (refuse).
Here's some examples:
- 선생님께 이 물건을 전해 달라는 부탁을 받았어요: I was asked to deliver this item to you.
- 미안하지만 그 부탁은 들어줄 수가 없구나: Sorry but I can't do what you're asking for you.
- 회사는 직원들의 요구를 받아들였다: The company accommodated employees' demand.
- 그는 내 요청[부탁]을 받아들였다 [수락했다]: He accepted my request.
- 그는 내 부탁을 들어줬어: He did for me what I asked of him.
- 그는 내 요청[부탁]을 거절했다: He declined my request.