On August 14th 1945, Japan surrendered. The surrender was announced to the world on the following day, with both the ceremony and the formal signing of surrender taking place on September 2nd 1945.
VJ Day is thus the day that marks the end of World War II and the end of fighting against Japan, and is referred to as either Victory over Japan Day or sometimes Victory in Japan Day.
There is a degree of confusion over what the actual date of VJ Day is. Each of the dates listed above could reasonably be believed to be VJ Day, with some of the confusion actually having been caused by none other than President Harry S. Truman himself.
The great majority of modern observances of VJ Day are on August 14th, the day when the government of Japan cabled its surrender to the United States. The news of the surrender was announced all over the world on the 15th, sparking spontaneous celebrations of the final ending of World War II. September 2nd is when Tokyo Bay played host to the formal surrender ceremony on the USS Missouri, chosen for the formal surrender ceremony due to the fact that she had been named after the home state of President Truman. President Truman declared at that time that September 2nd was VJ Day. One thing is certain: whichever date you believe was VJ Day, the war had finally come to an end.