That depends on one's worldview. As a student of history, I would have to pick August 6, 1945, the day the United States dropped the first atom bomb on Hiroshima, Japan. Suddenly, for the first time in world history, mankind now had the ability to totally destroy himself. This weapon, and the more powerful weapons which have followed it, now make another world war unthinkable--or a least horrifically unimaginable. And, until recently it seemed with the balance of power between the United States and the USSR, mankind just might prevent such a disaster. However, with the breakup of the Soviet Union and the rise of terrorism in the world, the nuclear scenario now once again seems possible if we do not find a way to rid ourselves of the nuclear threat.
On the other hand, I also consider June 7, 1967 to be a very important day, religiously. That was the day that Israel retook the city of Jerusalem. On a purely personal level, I consider that day to be the day that restarted the clock towards the second coming of Christ. In a way, both days, to me, seem to dovetail. Mankind enabled himself to destroy his kind on August 6, 1945 and God will come again to prevent that destruction.