"It wasn't the first trial dealing with the camps, nor was it one of the major ones. "

The major War Crimes Trials occurred immediately after the war, under the administration of Britain, France, the US and the USSR (known as the International Military Tribunal).  The most important were held in Nuremberg, and involved senior figures of the Nazi regime such as Hermann Göring, Rudolf Hess and Joachim von Ribbentrop.  Most were sentenced to execution, as were a number of concentration camp personnel such as Amon Göth and Irma Grese.  The witness testimonies were key to informing the rest of the world about what had taken place in the camps, although the majority of evidence came from the Nazi's own files which were now in Allied possession.  By the end of 1945, both East and West Germany were authorised to conduct the trials of their own citizens who had committed war

crimes or associated transgressions during the war, and this continued for several decades as evidence came to light and people were traced.  Even today, former Nazi guards or administrators are called upon to answer for their crimes.   

For a very good explanation of the process used at the Nuremberg trials, follow this link.