What does the kite symbolize in the relationships between Amir and Hassan, Amir and Baba, and Amir and Sohrab?
Good question. Kites are, of course, an important motif for the the novel, and they often represent a connection between two characters throughout the book.
1. Amir and Hassan: Kites (at first) make up the positive elements of Amir and Hassan's relationship. While Hassan loves hearing Amir's stories, and both boys greatly enjoy watching American movies together, kites are the only enjoyable activity that the boys participate in which forces them to be partners. Even though in kite flying, Amir still has the upper hand (he owns the kite and gets all the glory), he must rely on Hassan. In other activities, Amir doesn't need Hassan--he seems to tolerate him. Unfortunately, a kite eventually becomes a symbol of Hassan's worth to Amir. Amir is willing to betray his friend for a kite, symbolizing ultimate disloyalty.
2. Amir and Baba--the narrator specifically states what the kite represents between Baba and him. In Chapter 6, he acknowledges that:
"Baba and I lived in the same house, but in different spheres of existence. Kites were the one paper-thin slice of intersection between those two spheres" (49).
For Amir, the kite represents his only chance to win Baba's approval--something he wants more than anything. Amir is not athletic, does not share Baba's interests, and seems to be unlike his father in every way. The one area that Amir is gifted in--writing--does not impress Baba. Thus, Amir sees winning the kite tournament and bringing back the last fallen kite as the sole way to please his father.
3. Amir and Sohrab--the kite connection between these two characters is a reversal of the first. Amir wants desperately to help a physically and psychologically bruised Sohrab; so he takes the more servile position in kite flying and runs the kite for Sohrab. Sohrab is also the last "fallen kite." Just as Assef wanted the literal last kite earlier in the novel, he wants to claim Sohrab as his "victory prize." In each situation, Amir has an opportunity to fight for the last fallen kite and prove his honor. With Sohrab, he makes the right choice.