The main conflict is a man vs. self conflict. Amir lets something terrible happen to Hassan and spends the rest of his life trying to make up for it. For a long time he fights his guilt and tries to repress it, but then he finally has the opportunity to actually do something about his guilt. Though the larger question of whether or not he made his penance is left up to the reader, Amir eventually resolves his conflict with himself by refraining his guilt into a call to action to nurture Hassan's son.
Well, in my opinion "The Kite Runner" is about the loss of childhood innocence; the cruelty of a hard and unforgiving world ripping the naive eyes from the boys sockets. It's also very political, however, I tend to think of the political aspects as a subplot.