I tried to summarize this the best I could but it is a bit longer than a paragraph!
Starting in chapter 6 Bruno's family has moved from Berlin and is now in Auschwitz. Bruno is practically inconsolable that he has moved from Berlin away from his friends and former life. Over time Bruno becomes determined to find something new and entertaining to do in his new home. Before Bruno has any ideas Father hires a teacher to continue his children's education. The teacher loves history and sparks Bruno's love for adventure. He decides he will explore his backyard and the people he sees from the window in the pajamas.
Bruno follows the path to Auschwitz and comes across a boy names Shmuel. They boys have common interests and despite a fence separating them they begin to become friends. Bruno goes to the fence everyday after he is done with his schooling to meet his new friend Shmuel. He often brings him food when he does go. It is obvious though that Bruno does not understand the gravity of the situation as he complains to Shmuel that he would rather be on his side of the fence. He later reflects that Shmuel had told him his grandfather was missing and this made his remaining family very upset. Bruno begins to think he may have misspoken to his friend.
After betraying Shmuel when he works at his home as a servant and learning he will be moved back to Berlin because his parent's displeasure with how the children are being affected, Bruno is compelled to play with his friend. He meets Shmuel and climbs underneath the fence so that he could say the played together just once. He is also determined to help Shmuel find his father who is missing.
As the two boys run through the camp to find Shmuel's father they are rounded up by the soldiers and brought to what Bruno believes is shelter. Unfortunately, they are brought to be poisoned and killed. After not returning home there is a search for Bruno and after several months Bruno's father uses evidence to find out what happened. He is inconsolable and eventually is relieved of his command.