The contest for world leadership that began after world war 2?

The Cold War. The two superpowers that were in the world as of 1945, Soviet and the United States, were governed by two different styles of government, Communism and Democracy. The Soviets, wanting more countries to trade with (its normal trade being to democratic countries, who boycotted all trade with the U.S.S.R), wanted the government type of Communism to spread into other countries that were either democratic or republics. Countries that were democratic, particularly the United States, wanted to stop this. The Soviets began funding and arming Communist, forces in different countries. these led to the reign of Fidel Castro in Cuba, the Vietnam War, Korean War, and a Communist government in China, among others.