Woodrow Wilson was awarded the 1919 Nobel Peace Prize because of his efforts to create a just peace after WWI and, as part of that, to set up the League of Nations. This is somewhat ironic because the Treaty of Versailles and the League of Nations both failed. The treaty helped lead to WWII and the League was never very effective.
Wilson wanted WWI to be "the war to end all wars." He wanted to set up a system after the war that would ensure that countries would no longer go to war with one another. He had a famous plan, called The Fourteen Points, that was meant to make this happen. Wilson tried hard to get the Fourteen Points incorporated into the Treaty of Versailles. He was not altogether successful, though. He won the prize for these efforts, and particularly because he did succeed in getting the League of Nations (very similar to the United Nations today) set up.