The day the symbols of American imperialism (Twin Towers) and American militarism (Pentagon) were attacked; also the day most average American citizens realized there were real-life consequences for complacency and apathy in the face of criminal policies carried out by their government.
This was a significant moment in modern history because it marked the first time ever members of the oppressed nations actually fought back by striking at the center of world trade and finance. The World Trade Center and the Pentagon, both legitimate targets according to the very same standards of the US military, were attacked because the terrorists saw them as the main source of Western economic exploition and oppression of the Third World (through corporate globalization and military intervention).
Finally, it should be said that contrary to popular thought, terrorism is NOT a weapon of the weak. In fact, it is most often a weapon of the strong, and it has been practiced by powerful states for thousands of years, except they never call it "terrorism", but "liberation", "counter-terrorism", "defense", or some other self-justifying label. Examples include Rome, Nazi Germany, Israeli "Defense" Forces, and the US State Department.
..A captured pirate was brought before Alexander the Great. "How dare you molest the sea?" asked Alexander. "How dare you molest the whole world?" the pirate replied, and continued: "Because I do it with a little ship only, I am called a thief; you, doing it with a great navy, are called an emperor."