According to the Oxford dictionary, a total war is defined as follows:
A war that is unrestricted in terms of the weapons used, the territory or combatants involved, or the objectives pursued, especially one in which the laws of war are disregarded.
With this definition in mind, we can easily see how World War II was a total war. It was a winner-take-all contest. Whichever side lost would be forced into a total, unconditional surrender. It was not a limited war with limited objectives that used limited firepower to achieve a goal. Instead, the goal was the total crushing of the enemy.
Neither side held back on the weaponry they used, and in fact, both sides battled to develop ever more lethal weapons. The Nazis developed V-2 rockets that could be launched from the ground and carry explosives long distances. The U.S. developed the atomic bomb, which delivered a stunning and unprecedented level of destruction.
The war also did not make distinctions between combatants and non-combatants. The Nazis killed many civilians in Poland and the Soviet Union to try to clear the space for Germans, and did not care how many civilians their blitzkrieg against England took out. Likewise, the Allies used air warfare, including firebombs, against cities largely built of wood. They pulverized these cities with no concern for civilian deaths, wanting to break the will of the enemy. Another disregard of non-combatant zones took place when the U.S. atom bombed Hiroshima and Nagasaki in order to avoid ground combat in Japan (which may have taken as many civilian lives).
The fire bombings of civilians were considered a disregard of the rules of war, as were the Nazi's mass execution of Soviet prisoners of war near the war's end. Finally, the war wasn't over until the Allies achieved the total defeat of Germany and Japan.