There were two major significances of these trials for the world as a whole.
First, it helped to bring closure to World War II. The Nazis had committed some of the worst atrocities in the history of the world and it was important that they be brought to justice for it. By holding these trials, the Allies were able to give people some sense that justice had been done and the world could proceed from there.
Second, it helped to establish the idea that there are moral laws that all people (even if they are in government and even in time of war) must obey. This idea has not been universally accepted and has not prevented further horrors from being committed. But it has changed what we think of as acceptable behavior for those in power. And it has had some effect, a fact which can be seen in the trials of people from the wars in the former Yugoslavia and the recent conviction of Charles Taylor (former ruler of Liberia, convicted of supporting rebels who committed war crimes in Sierra Leone).