-아/어 주다
a/eo juda
Handing the action over: -아/어 주다
The picture The action you perform sliding across and landing in the other person's hands. 기다려 주세요 makes the waiting cross over to me; 기다리세요 simply tells the other person to wait.
Where the picture shows up in -아/어 주다
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an action coming toward me
잠시만 기다려 주세요.
jam-si-man gi-da-ryeo ju-se-yo
Please wait a moment.
The other person standing still until your work is done, setting their time down in front of you
기다리세요 is an order; 기다려 주세요 is a favour asked. With customers it is always the second.
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asking carefully
사진 좀 찍어 주실 수 있어요?
sa-jin jom jji-geo ju-sil su i-sseo-yo
Could you take a photo for us?
Holding out your camera with both hands, and they press the shutter and hand the photo back to you
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an action going up
제가 도와 드릴게요.
je-ga do-wa deu-ril-ge-yo
Let me help you with that.
Taking the heavy box out of your senior's hands and carrying it over in his place
When the receiver outranks you, 주다 becomes 드리다.
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a kindness that reached me
친구가 공항까지 태워 줬어요.
chin-gu-ga gong-hang-kka-ji tae-wo jwo-sseo-yo
A friend drove me to the airport.
Riding in a friend's passenger seat to the airport, and he lifts your bag down at the curb too
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asking a close friend
문 좀 닫아 줄래?
mun jom da-da jul-lae
Can you close the door for me?
You point at the door with your chin from the sofa, and your friend gets up and shuts it
All of it in one picture
-아/어 주다 is an arrow: what leaves your hands lands in someone else's. Someone's waiting set down in front of you becomes a kindness; a shutter pressed and the photo handed back is a request fulfilled. Take the box and the arrow points up toward your senior; get driven to the airport and the arrow points down to you; a friend shutting the door is the same arrow in casual speech. So a sentence with -아/어 주다 always records whose this action was for.