Others have taken a political bent in answering this question, which is completely warranted and justifiable, but I would also offer that artistic movements fundamentally changed the shape of our nation.
As bullgaitor noted above, Vietnam changed the nation. By 1969, many young people had had it with the more than a decade-long war. They declared the "Summer of Love," in 1967 and Woodstock, still the greatest open air music festival ever, occurred in upstate New York in 1969. The protests of young people against the war and the deaths of soldiers changed the course of the conflict.
In art, Andy Warhol and his Factory were influencing the commerical art world with his now-iconic paintings of the Campbell Soup Cans and his portrait of Marilyn Monroe. The works were like nothing before and have spawned countless imitators.
In theater, Tom Stoppard's existentialist drama Rosencrantz and Guildenstern are Dead reflected the confusion of events in a world where people do not have direct knowledge and try to make sense from a jumble of nonsense...quite analogous to America's feelings about the war.
I would argue that not since the Harlem Renaissance, also a reflection of dissatisfaction with the course of the nation, has a decade so served to make its mark on the Arts and on our nation's history as well as our future.
It would be hard not to think of the Vietnam War as the defining historical event of the 1960s. Although the first American advisors were first sent during the Eisenhower Administration during the late 1950s, American involvement escalated during the John F. Kennedy years and, later, under his successor, Lyndon B. Johnson. The war served as the prime reason Johnson chose not to run for re-election in 1968, since he understood that many Americans had lost confidence in him because of Vietnam. Richard Nixon's presidency fared no better, and when the last American troops were brought home, they became the first serviceman to return from overseas without a clear victory. It was a war that involved nearly every American in some way or another, and nearly every person in the United States knew someone serving there. Unlike the two World Wars and even the Korean War, the War in Vietnam was highly unpopular with most Americans as well as world-wide, and American involvement was highly criticized in most circles. It was a different type of war, and American military strategy was flawed in many respects. The American populace also hated the military and political secrecy and misinformation that dominated most reports; and even more unpopular was the draft, which forced thousands of young men to serve in a war in which they did not believe. In the end, the U. S. shamefully left South Vietnam to fend for itself, and the inevitable communist takeover soon became a reality.
There are so many events from the 1960s that could be said to "mark the history" of the US. I'll mention three, but there are many more.
First, I would point to the Cuban Missile Crisis. This was a major event in world history and it symbolizes the fact that the Cold War was a major part of the '60s. The Gulf of Tonkin Resolution could be used in this way as well.
Second, I would mention the March on Washington and Dr. King's "I Have a Dream" speech. This symbolizes the importance of the civil rights movement during this decade.
Finally, I would say we should look at the 1968 election. We should look at the Democratic convention in Chicago as an example of the strife that tore the country apart during that time. The election of Nixon also marks the turn towards conservatism that came with the end of the '60s.