Although it is hard to choose one conflict in this story--there are so many--I would say the main conflict is one of freedom vs. confinement. It would be very simple to say, "Yes, Bruno, as a young German boy, is free, and Shmuel, as a young Jewish boy, is not," yet that is oversimplistic. We can certainly understand how it is that Bruno is free: he lives in a big house, is free to come and go as he likes, he can eat whatever and whenever he needs to, he can wear whatever clothes he chooses, and he has friends with whom he is free to play. And it's easy to see how Shmuel is confined, having been first taken from his home and confined to a single room, then taken from that room and locked up in a train, then taken to the camp and forced into striped pajamas that are his only covering (and that look like prison bars), and then imprisoned, literally, in Auschwitz. Yet Bruno is not as free as he appears, and in this, he lives a loose parallel with Shmuel. Bruno is forced to move to a place he does not want to go, where he has no friends, and from which he cannot leave. He cannot leave to return to his previous home or his friends, and neither can he leave the grounds of the new house. In another loose parallel to Shmuel's confinement, Bruno has a Nazi soldier charged with making sure he does not.
We can see the freedom vs. confinement conflict play out in another scene when Shmuel is part of a work detail brought to Bruno's home to clean glassware for an upcoming party. While Bruno is free to roam the house, Shmuel is not. But more than that, at this point, Bruno and Shmuel have forged a friendship, in which they both have found a freedom from their loneliness and isolation. In the freedom of this friendship, Bruno offers his hungry friend a piece of chicken and Shmuel accepts. When Kotler shows up accusing Shmuel of stealing it, Shmuel, feeling free in his friendship with Bruno, says that Bruno is his friend and gave it to him. But Bruno, faced with Kotler's anger, is not free to answer that truthfully and denies it, which leads to Shmuel's beating. So although their friendship has provided some freedom from their loneliness, both are confined by their roles in it.
The end of the novel brings one last significant clash between freedom and confinement when Bruno, whose own father has been 'missing' on a trip to Berlin, sheds his clothes to don a borrowed pair of striped pajamas and wriggles under the fence into Auschwitz to help Shmuel look for his missing father. Here we see a literal move from freedom to confinement for Bruno, and as he realizes just how bad things are for Shmuel, he longs to go home. Yet he becomes a prisoner to his sense of loyalty, to the friendship that has brought him some small measure of freedom from his loneliness, and his confinement is fully realized when the running boys are surrounded by armed soldiers and marched into the gas chamber. Although their innocence in a sense momentarily frees them from the reality of their situation, they are both about to die as prisoners. In a fitting end, Bruno's father is forced to give up his freedom when the Allied soldiers arrive at the camp.