The conflict between Hassan and Amir arises when Amir decided to run away instead of helping Hassan to escape this harsh treatment from Assef. This incident has left a deep scar inside Amir's heart from the enormous guilt that he had towards Hassan.
Amir felt sad when he saw how Baba treat Hassan so well when he desperately craves for Baba's attention. In this case, Amir tried to fake his interest that he loves to play and watch soccer just to catch Baba's attention.
Assef and his gang saw Amir and Hassan together and they corners them. After that, Assef took out his brass knuckle tried to hit them and claimed that Hassan does not deserves to be Amir's friend because he is a Hazara while Amir is a Pashtun, which is an unacceptable to him.