This incident is a turning point in the life of Amir, Hassan, Baba, and Ali. As a result of what happens in an alley one day, all four of their lives are irrevocably changed.
On the day in question, Amir and Hassan are in the kite competition that occurs each year in Afghanistan. Amir is determined to win this competition to prove to Baba that he is worthy. Hassan, who is his devoted friend (and servant), is equally determined to help Amir win.
Once Amir has cut down the last competing kite, he is the winner. But there is a tradition of retrieving the last kite that has been cut down, and this is where Hassan's great skills come in. He is always able to sense where a kite will land and retrieve it. That is why he is called "the kite runner."
When Hassan goes to retrieve the kite, he finds himself in a grim alley, surrounded by the local bully, Assef, and the bully's friends. In the meantime, Amir has followed Hassan, and when he realizes that Hassan is trapped with the kite, being threatened by Assef and his friends, he hides and witnesses Hassan being sodomized by Assef.
The reason this incident is central to the story is because, while Hassan had defended Amir from Assef in the past, Amir does nothing but hide when Assef does this terrible deed, and the rest of the story flows from Amir's weakness. After, he pretends he is unaware of what has happened, but he is haunted by guilt and cannot bear being with Amir. He manipulates events so that Amir and Ali are forced to leave the household, by making it look as though Hassan has stolen from him. There are, of course, many more consequences that flow from that day, but I think it is safe to say that Amir spends the rest of the book trying to make up for his failure on that day.
As the book opens, if you recall, Amir is told by Rahim Khan that he has a chance to "be good again" (2). That is a chance for Amir to repair the damage he had done as a child on that day, a chance at redemption, which is one of the most powerful themes of the book.