The relationship between Amir and Hassan is crucial to the themes of this novel because it is this friendship and the changes it undergoes that illustrate betrayal, regret (guilt) and forgiveness.
Although they do not know they are really biological half brothers, Hassan and Amir grow up together and as very young children, are as close as brothers. The big difference is, however, that Hassan is living like a servant in Amir's house and as such, he shows the proper deference to Amir as his better, also because Hassan is of a lower religious sect in Afghanistan. When they are very young children, this difference is not important because neither really knows what it means due to their childhood innocence. However, as the boys grow older, they begin to see that there are cultural and religious differences that separate them.
Amir betrays Hassan's friendship by failing to stick up for him or go for help when Hassan is raped. After this event, their friendship is changed because Hassan continues to befriend Amir, but Amir feels guilty because of his cowardice and pulls away from Hassan. This leads to regret on the part of both boys as they grow into men, only on different levels. Amir further betrays Hassan by not sticking up for him when Hassan is falsely accused of stealing.
Hassan is killed by the Taliban before he can forgive Amir, but Amir seeks atonement and forgiveness by trying to adopt Hassan's son. His regret never leaves him, however. Amir's suffering and maturity do help him understand and ultimately forgive his father for not telling him the truth about Hassan and for raising him the way he did both in Afghanistan and in the United States. Unfortunately, Amir does not learn the truth from his father.
You can read about the themes here on eNotes.