The exposition of any story is the introduction, in which the setting and main characters are detailed. The exposition ends with the inciting action, an event which triggers the conflict which changes everything in the story.
In The Boy in the Striped Pajamas by John Boyne, the exposition begins with nine-year-old Bruno's discovery that he and his family will be moving. He arrives home to find that the family's maid is packing up all of his belongings, and at first he is afraid he has done something bad and will be sent away from his family (though he would be perfectly okay if his sister had to leave). We learn about the house Bruno's family will be leaving in Berlin, a place which Bruno likes to explore. We learn that Bruno is small, even for a nine-year-old, and has a rather contentious (but perfectly normal) relationship with his twelve-year-old sister, Gretel. We learn that Bruno has three best buddies and says he is devastated to have to leave them. We learn that Bruno's mother does not want to move any more than he does, but she is the adult and will do what she must. We learn that Bruno's parents set a lot of rules for proper behavior and expect their children to follow them.
As much as anything else in the opening chapter (the family arrives in their new home in chapter two), we learn that this story is set in World War II in Hitler's Germany. When Bruno complains about having to change everything, Bruno's mother reminds him that he has been complaining that everything has changed so much here in Berlin, too. He says,
"Well, I don't like the way we have to turn all the lights off at night now."
Obviously the setting is Germany; since the war is now coming to them (which is why they have to have blackouts each evening) we can surmise that this is toward the end of the war. Things in Berlin are changing, but Bruno and his family are leaving and will not experience the changes there; they will be in their new home.