Chapter seven of The Kite Runner by Khaled Hosseini is a pivotal chapter in the novel, the point in time when questions are answered but everything changes. Amir is the narrator and protagonist of the novel, and he is twelve years old in this chapter. Ironically, what happens in this chapter is both his proudest moment, as you say, but also his most shameful.
This chapter is full of flashbacks, dreams, and remembrances, but the primary setting is an event, the kite-running tournament. Amir knows that his father, Baba, is expecting his son to win, just as Baba used to do.
[T]his was my one chance to become someone who was looked at, not seen, listened to, not heard. If there was a God, He’d guide the winds, let them blow for me so that, with a tug of my string, I’d cut loose my pain, my longing. I’d endured too much, come too far. And suddenly, just like that, hope became knowledge. I was going to win. It was just a matter of when.
The competition begins with over fifty kites, but eventually it comes down to just two: Amir's kite and a blue kite. Hassan is Amir's kite runner, and he has a special gift for finding the final kite. When the blue kite is finally cut, Hassan takes off to get it, and Amir anticipates how much Baba is going to love Amir for winning the contest.
In the end Amir has his proudest moment when Baba congratulates him for winning; however, Amir wins at the expense of Hassan. Hassan was raped by a cruel, cowardly bully; Amir saw it happen and did nothing to stop it. Hassan will only get the kite if Amir ignores what he sees, and that is exactly what he does. Even worse, he does not speak one word to his bleeding and broken friend after the assault. Some part of Amir knows what he is doing is wrong, but it is not strong enough to override Amir's desire to win his father's approval--something which is very difficult to get. Hassan is the sacrifice Amir is willing to make, and Hassan is willing to sacrifice himself for Amir.
This winning is the first time Amir has ever felt as if his father is even a little proud of his son; however, this winning is also the source of lifelong guilt and shame for Amir.
It happened in just the way I’d imagined. I opened the door to the smoky study and stepped in. Baba and Rahim Khan were drinking tea and listening to the news crackling on the radio. Their heads turned. Then a smile played on my father’s lips. He opened his arms. I put the kite down and walked into his thick hairy arms. I buried my face in the warmth of his chest and wept. Baba held me close to him, rocking me back and forth. In his arms, I forgot what I’d done. And that was good.
Though both boys win the tournament and have every right to feel proud of their accomplishment, Amir's and Hassan's lives are forever changed by this episode and Amir is haunted for most of his adult life because of it.