There are actually a number of ways to answer this question, but the conflict that we have the most insight into is Amir's internal conflict. This internal conflict is with him his entire life, in part because he feels he is to blame for his mother's death in childbirth and for his father's indifference toward him. However, the inner struggle intensifies after the kite-fighting tournament. Because it is so important to him to win the tournament, Amir watches his friend Hassan be sexually assaulted (a bully is trying to take the kite from him, even though Hassan won the kite fairly) and doesn't try to intervene or get help. Amir's decision to flee the scene has lifelong ramifications, as he struggles to overcome the guilt associated with his weakness and betrayal. Amir takes more overt actions against Hassan, eventually resulting in Hassan and his father Ali leaving the house (they are servants to Baba and Amir). However, his acting out and his eventual move to California with Baba do not give Amir the solace he seeks. It is only when he goes back to Afghanistan and rescues Hassan's son Sohrab from the Taliban and brings him home as his adopted son that Amir begins to mend. He finally feels he is atoning for his sins, so he is working toward resolving his internal conflict.
The novel also presents conflicts in the relationship between Amir and Baba and in that of Amir, Hassan, and the bully Assef. The larger context of the novel presents the conflicts between racial, ethnic, and socioeconomic groups in Afghanistan. Hassan's status as Hazara makes him inferior to Amir and Assef in the social hierarchy, and these differences result in violence and verbal abuse. The racial and ethnic conflicts in Afghanistan certainly inform the personal conflicts between characters as well as Amir's internal conflict, as he must wrestle with the notion that he actually thought of Hassan as lesser than himself.
The main conflict in the book is the tension between the elite and the disenfranchised in Afghanistan. Amir is the legitimate son of his father, Baba, and enjoys a life of wealth and privilege. His friend (who turns out to be his half-brother) is Hassan, a boy born into a less privileged ethnic and social class. Amir is Pashtun, a member of the more elite ethnic group in Afghanistan, while Hassan comes from the Hazaras, whose members are treated as outcasts. Though there are many parallels between Amir and Hassan, Amir lives a life in which he is largely spared of hardship. Hassan, on the other hand, is raped for having tried to rescue Amir's kite, and, later, Amir blames Hassan for stealing money that Amir plants on him. It is only many years later, when Amir adopts Hassan's orphaned son, that Amir tries to make up for the wrongs he has committed as a member of the privileged class against his friend and half-brother, who is an illegitimate son and the member of a culture whose members are treated as pariahs.
One could argue the main conflict in the story is Man vs. Self as Amir struggles to atone for his past sins as an adolescent growing up in Kabul. As an adolescent, Amir struggles to please his father, which adversely affects his self-esteem and confidence. Amir's inability to make his father proud creates a sense of envy towards his Hazara friend, Hassan. Amir's lack of confidence and esteem manifests itself into his feelings of jealousy towards Hassan.
On the day Amir and Hassan win the kite-flying tournament, Hassan runs after the blue kite to finish the game without Amir. Amir ends up searching for Hassan and finds him in a dead-end alley surrounded by Assef and his two friends. Instead of defending Hassan and sticking up for his friend, Amir hides and witnesses Assef rape Hassan. After witnessing Hassan get raped, Amir suffers from extreme guilt and can no longer stand being in the same home with Hassan. Amir ends up forcing Hassan to leave Baba's estate and flees to America with his father, where he represses his memory of Hassan.
As an older man, Amir's repressed memories and guilt begin to haunt him. Amir becomes an insomniac and fortunately gets a call from Rahim Khan saying, "There is a way to be good again" (Hosseini, 2). Amir realizes that he finally has an opportunity to atone for his past sins and find redemption after having an enlightening conversation with Rahim Khan in Pakistan. For Amir to find redemption, he travels to war-torn Kabul, fights Assef, and ends up adopting Hassan's son, Sohrab. By the end of the novel, Amir atones for his past sins, by risking his life to save Sohrab, and finds redemption.