Rahim Khan is a figure from Amir's (the narrator's) childhood in Afghanistan. When the novel begins, Rahim Khan calls Amir and tells him to come back to the Middle East, where he will have "a way to be good again." After this phone call, Amir goes back and narrates the story of his childhood in Afghanistan with his friend/servant Hassan and the tragic end of their relationship. In this story, we learn more about who Rahim Khan is and what impact he has on Amir's childhood.
Rahim Khan is the business associate and best friend of Amir's father, Baba. Amir remembers Rahim Khan gathering in Baba's home with other friends and businessmen. During these gatherings, Amir always felt ignored by his father, but Rahim Khan takes the time to talk to Amir and support his interests. While Baba feels like he and Amir have nothing in common, Rahim Khan gives Amir the tools and encouragement he needs to pursue his dream: becoming a writer. When Amir is an adult, we find that he has published a novel. It may not have been possible for Amir to stay committed to his dream if not for Rahim Khan's support.
In the present day of the novel, Rahim Khan provides Amir with an opportunity for redemption. As a child, Amir betrayed Hassan by watching him be assaulted and doing nothing to stop it. Later, Amir lied to get Hassan and his father Ali removed from Baba's home. Amir and Hassan never saw each other again, and though readers know Amir feels guilty about what he did in the past, he really has no chance to be forgiven or to make amends with Hassan because as he finds out once he returns to Pakistan to see Rahim Khan, Hassan was killed by the Taliban. Rahim also reveals a crucial piece of information to Amir: Hassan was his half-brother, as Baba also fathered Hassan. Rahim Khan entrusts Amir with a mission: to retrieve Hassan's son Sohrab from an orphanage in Afghanistan and bring him to Pakistan to be adopted by the Caldwells, an American couple. Amir fulfills his duty but later finds out that there were not Caldwells; Rahim had intended for Amir to form a relationship with Sohrab and adopt him. Despite his immediate reluctance and the hardships he must undertake along the way, Amir does adopt Sohrab, brings him back to the U.S., and defends him against ethnic prejudice (in a way that he never had the courage to defend Hassan during their childhood). It is Rahim Khan's understanding of Amir's character that sparks Amir's quest for redemption.