What exactly does The Kite Runner say about issues of social class, and how does this relate to us in society today?
The issues of social class are central to The Kite Runner, especially concerning Amir and Hassan, and to a lesser extent Baba and Ali as well. It is the core of all problems and suffering in the novel, with the exception of simple human cruelty, which transcends all social class. It is worth noting, however, that other things transcend those lines as well but are less obvious, at least to societies in large. For example, the reader notices quite early on that Ali and his son Hassan are portrayed in a more positive light than Baba and Amir. This contrast only gets sharper as the action progresses, especially concerning Amir, who has to spend the second half of the book making up for mistakes he made in the first.
This isn't to say that people in power are always worse than those who are oppressed, but it shows that the lines social class draws are completely fictional and arbitrary. History has shown us time and time again that every class from kings to slaves births heroes and villains all the same. What the social class does account for is opportunity, and that remains as relevant as ever.
Today's society has all those same issues, and they are likely to remain ever-present. As human beings, it is very natural to us to think of ourselves in terms of "us" and "them," whether the "us" is our family, our religion, our nationality, or something else. In itself, it is nothing negative. The problems begin when people start to think that belonging to one group makes them better than the people in another group.
In the case of The Kite Runner, as the other answers have already pointed out, the characters don't really raise the issue of whether the Pashtuns are somehow actually objectively better than Hazaras. They live in a society where that is already accepted as a given. It means that while Hassan is—from the reader's point of view, at least—objectively a better person than Amir, it does not matter. He will still be persecuted based on his ethnicity; he will still be denied proper education and opportunities.
In addition, the other issue with social class is the continued perceived superiority of the Pashtuns. It is said that absolute power is corruptive, and it shows in the novel in the actions of the Pashtuns. Even good men like Baba turn a blind eye to the shameful treatment of Hazaras, because the unpleasant truth is that people who enjoy luxuries and success are reluctant to give it up for the sake of those who do not. In conclusion, The Kite Runner realistically portrays that there are good and bad people all over the world and in every social class. In the end, however, a person's worth is only defined by their own actions.