What is the history of concentration camps in Nazi Germany during the time of The Boy in the Striped Pajamas by John Boyne?

Because you are referring to The Boy in the Striped Pajamas, I assume you are speaking of the specific history behind the Nazi concentration camps before and during World War II.  (Of course, there have been other internment camps both before and after this time period.)   In short, Nazi concentration camps lasted from 1933 to 1945.

The first concentration camps appeared immediately after Hitler became chancellor of the Nazi Party in Germany.  The treatment at the concentration camps became particularly harsh after Heinrich Himmler and his SS took complete control of them.  The population interned at the camps escalated rapidly, eventually numbering in the millions.  The largest population set “concentrated” here was the Jewish population (specifically from Poland).  The second largest population set “concentrated” here was the Soviet prisoners of war.  However, there were many other people placed in the camps as well, such as Catholic priests, homosexuals, intellectuals opposing the Nazi regime, etc.

As you can see from the reference above, the term “concentration camp” is a reference to “concentrating” a certain undesirable part of the population in a small area.  The Jews in Poland, then, were confined to “ghettos.”  As time went on, concentration camps became synonymous with extermination camps because many people “concentrated” within them were tortured and killed, often in mass numbers.  The treatment was horrible, with prisoners being subjected to experimentation, starvation, overwork, and execution.  One of the most common methods of murder within the concentration camp was the gas chamber.  What looked like an enormous shower room was actually a dispensary for lethal gas that killed hundreds of people at a time.  The bodies were then deposited in mass graves.

The history behind Nazi concentration camps is a grim history.  With close to 43,000 Jewish ghettos and over 20 million people killed, the history of Nazi concentration camps is a history of evil in the attempt to exterminate entire populations.