The accepted date for the beginning of WWII is September 1, 1939, when Hitler's armies invaded Poland. Britain and France threatened war if Germany did not leave the conquered territory; within a week, Britain and France declared war on Germany. WWII would not officially end until September 2, 1945, when Japan surrendered to the Allied forces on board the U.S.S. Missouri.
While this is the accepted date for the beginning of the war, there are other dates that should be considered signposts on the road to true "world war." In September of 1931, Japan invaded Manchuria. While the League of Nations and the United States denounced this act of aggression, Japan was not threatened with force. In 1936, Mussolini invaded Ethiopia. In 1938 and 1939 Hitler annexed Austria and Czechoslovakia, respectively. In all of these invasions and annexations, the world community was desperate to avoid war, so it did not even threaten the use of force.