I SUGGEST WIKIPEDIA!
But anyway.
The war began de facto on September 1st, 1939 with the Nazi German invasion of Poland. However, hostilities began long before this with the Italians in Africa and the Japanese in China and Korea. The Italian-Japanese-German alliance was the core group of the Axis Powers (and was nicknamed the "RoBerTo" alliance because of the capitals of these nations).
Germany was controlled by the Nazi Party (Nationalsozialistische Deutsche Arbeiterpartei), a fascist socialist group that was headed by the Austrian Adolf Hitler, a WW1 vetran who spent a time in jail and published Mein Kampf. Italy was run by Mussolini, another fascist, and Japan by Emperor Hirohito, but they didn't cause as much trouble as Hitler.
Before the war, Germany had previously annexed Austria and Czechoslovakia. England and France, trying to appease Hitler, ultimately do nothing.
Within 3 weeks, Germany conquered Poland (with the help of the Soviet Union per the Moltov-Ribbentrop pact). Once again, England and France do nothing. In another 2 months, Denmark, Norway, France, the Netherlands, Luxembourg, and Belgium. 
Germany continues conquering nations across Europe until June 1941, when he thinks it would be totally awesome to try and take over Russia. This is, ultimately, the worst move that anyone can make in any war ever. The Russians are pissed and retaliate, switching over to the Allied side. Later that year on December 7th, the Japanese bomb Pearl Harbor. The next day the US declares war on Japan, Germany declares war on the US, so the US declares war on Germany and Italy. The war is no moved to a fully global stage.
While all this is going on, the Germans proceed with the mass extermination of Jews, Gypsies, Poles, POWs, communists, Catholics, mentally retarded, homosexuals etc. across Europe in death camps. All in all, over 10 million people die in these camps.
Italy changes sides during the war, the Germans get their asses kicked at Stalingrad, the US and the British proceed to bomb the life out of everyone, the Poles do some uprising, and Vichy France continues to exist. Between 1942-3, the tide changes.
In 1945, the Allies win the war in Europe in May and in Japan in August after two atomic bombs are dropped on Hiroshima and Nagasaki. The war then allowed Russia to take over much of Eastern and Central Europe, setting the stage for the Cold War to follow.
If you have any questions, don't hesitate to ask me as I am rather well-versed in WW2 history.