As a lot of people have said, that day really did change the world.
But the specific moment that changed the world was the second plane hitting the South Tower. For 15 minutes we all thought there had just been a nasty accident. There's a video on YouTube of three guys filming the towers from their apartment about 5 or 10 blocks away, and as soon as the second plane hits a guy runs up behind them and starts screaming "That's terrorists! That's fucking terrorists, bro!" There's no better way to sum up that sinking feeling.
A fascinating and deeply troubling day in modern history.