What is an important quote said by Amir in The Kite Runner?

One important quote from Amir in The Kite Runner is the following:

I always felt like Baba hated me a little. And why not? After all, I had killed his beloved wife, his beautiful princess, hadn’t I?

What’s important about this quotation is that it helps us to understand the fraught relationship between Amir and his father, Baba. Amir has always felt that Baba resents him for causing the death of Baba’s late wife and Amir’s mother, who died in childbirth. And so, Baba has never been able to show the necessary affection toward his son, preferring to show it instead to Hassan, his illegitimate son.

There is an element of hypocrisy here, as Baba cheated on his “beloved wife, his beautiful princess”; so perhaps, in treating Amir as less than a son, he’s transferring some of the guilt he feels for committing adultery.

Because Amir feels that his father hates him a little, he’s all the more determined to win his approval, which he attempts to do by winning the local kite-flying tournament. If he cannot have Baba’s affection naturally, then he can at least earn it.

What quote represents Amir's downfall in The Kite Runner by Khaled Hosseini?

The Kite Runner, by Khaled Hossein, is narrated by the protagonist, Amir. Amir is raised by his father in Afghanistan, and it is fair to say he lived a relatively privileged life until the arrival of the Taliban and the beginning of the war. He and his father are forced to leave the country with virtually nothing; they go to America and have to work hard to be successful. It is logical to assume that the primary crisis in Amir's life was having to leave his former life behind; however, the beginning of what you refer to as his "downfall" happened several years before that, when Amir was twelve years old.

This novel begins at the end, so to speak, which means the opening lines are spoken by Amir years after his life changed and after he has had time to reflect on where and when everything went wrong for him. 

“I became what I am today at the age of twelve, on a frigid overcast day in the winter of 1975. I remember the precise moment, crouching behind a crumbling mud wall, peeking into the alley near the frozen creek. That was a long time ago, but it’s wrong what they say about the past, I’ve learned, about how you can bury it. Because the past claws its way out. Looking back now, I realize I have been peeking into that deserted alley for the last twenty-six years.” 

Though these words are cryptic, they are true. It takes several chapters for the story to unfold and for Amir to reveal the specific incident to which he refers, but it is this moment which irrevocably changes and shapes the rest of his life. “It may be unfair, but what happens in a few days, sometimes even a single day, can change the course of a whole lifetime....” 

When Amir's best friend (and his half-brother, Amir learns later) Hassan is being brutally treated by some cruel boys in an alley, Amir sees but does nothing. Hassan is aware that Amir knows what happened, but Hassan says nothing. His silence compounds Amir's guilt, and Amir eventually destroy's Hassan's life because of that guilt and jealousy over their shared father. 

While Hassan was the victim on that day so long ago, Amir was the one who suffered the consequences for of his own cruel, selfish behavior that day. There is one hope for redemption, offered to him by a family friend: “There is a way to be good again," he tells Amir; the novel ends with some hope after Amir chooses to take advantage of that opportunity to make things right.