What it means:
First things first: 주다 is a verb that means “to give”.
So -아주다 / -어주다 is a construction that gets added to a verb to create a compound verb. It creates the meaning that the verb is being done “for” someone, or “as a favour to” someone. It gives the nuance that the verb is being done for the benefit of whoever it’s being done to/for.
Because of this nuance that -아주다 / -어주다 creates, it is also often used in imperative sentences (sentences where you tell someone to do something) to soften the request. It would sort of turn a sentence like, “Shut that door.” into a request: “Please shut that door for me.”
How to attach it:
Follow the standard rules for attaching -아\어 particles. Read about it here. This grammatical principle gets used a lot, so we’ll do a lot of examples.
Boy With Luv:
네 모든 걸 다 가르쳐줘. ~ “Teach me everything about you.”
Original verb: 가르치다 – to teach
Make It Right:
그때처럼 날 어루만져줘. ~ “Caress me like you did then.”
Original verb: 어루만지다 – to caress
날 살려줘. 다시 기회를 줘. ~ “Save me. Give me a chance again.”
Original verb: 살리다 – to save someone’s life
절대 거부감 없이 날 받아줬던 너 ~ “You who took me back with no resistance at all.”
Original verb: 받다 – to receive
forever rain (by RM):
누가 나 대신에 좀 울어줬으면 해서 ~ “Because I’d like someone else to cry in my stead for a bit.”
Original verb: 울다 – to cry
나를 믿어줬던 모든 사람들을 배신하게 될까봐 ~ “I’m worried that I’ll betray all the people who’ve believed in me.”
Original verb: 믿다 – to trust, to believe in