In Chapter Nine, Amir uses the wristwatch he gets for his birhtday to frame Hassan. He hides it, along with some cash, under Hassan's mattress, then complains that it is missing. When it is found Hassan is naturally presumed to be the thief. His father, Baba's faithful servant, chooses to leave the household rather than bear the shame of such incrimination. From this point on, Amir and Hassan's destinies are forged in different directions. The time of their complicity and friendship is over.
It is only until much later that Hassan learns that Hasssan's real father is Baba, and that Hassan is indeed is half-brother. This moment of awakening comes too late to make ammends to Hassan (who has been killed), but Amir goes to great lengths to save his son Sohrab from Assef's clutches.
The wristwatch taken as a universal symbol evidently represents time, and with that time running out. More specifically in the context of the story, it represents Amir's act which once done cannot be undone, as there is no possibility of going back. For Amir the wristwatch also represents his privileged status as a legitimite son but also treachery. For Hassan, it represents discrimination, favouritism, and Amir's treachery.