I am Hispanic and one of the events that occurred in the 1960s that could have shaped my fate is, of course, the civil rights movement. Back in the day the Hispanic population was not as spread as it is today, and most of the rights that were discussed pertained mostly to black Americans.
However the fact that "Civil Rights" are rights for every civilian, these rights ended up involving immigrants such as myself, who came to America to make a better life through hard work in order to give back to the community.
The Vietnam "Conflict" in which the US involved itself in a civil war of another country killed several of my classmates. Each time I attend a class reunion, I am grieved by the loss of these friends who were cut down at 18 or 19 for no reason other than the evil of such men as Robert MacNamara and Lyndon B. Johnson.
The United States had not been involved in the Vietnam War my father may not have gone to college (to defer his draft status for a few years) and he may never have met my mom. While this is a personal example, I think that Vietnam has many long-reaching ramifications on the American Experience. It was the first war that was actively protested -- setting the precedent for protests about our current war. It was the baby boomer generation's first real challenge to their way of life or life as it had been defined for them.
The Civil Rights movement most certainly had an impact on how we (all races) interact with one another. I especially like the responder who said that our classrooms would be different if the Civil Rights movement has not happened. I work with a very diverse group of children and seeing them interact the way they do is very inspiring. Even thought they are different colors, have different cultures, etc. they treat one another as equals.
For me, the feminist movement (which many link to the publishing of Betty Friedan's The feminine mystique in 1963) has affected me the most as a woman. The push for equal pay for equal work and the encouragement to shatter the glass ceiling for woman in the workplace allowed me into many areas of the public sector that I probably would have had a hard time getting into without the hard work of brave women before me (IE: Journalism, etc.)
I do not think that anyone can deny the importance of the Civil Rights movement. While it passed before my time, I can sit in my classroom today and look at the faces of many different ethnicity. If the act would not have passed, I would not be able to do that.
Another thing that I think I would have lost, and many of you literature lovers, is the amazing writings by, well, everyone not white. While they still may have written, their works would have probably been pushed to the side and ignored. We would surely live in a literary place which lacked diversity.
I am a teacher. In my classroom, girls and boys of different races interact as if there is no difference between them. This would not have been possible if the Civil Rights Movement had not changed how people see race.
My mom was diagnosed with breast cancer in the late 1960s, and she fought a brave battle, undergoing a mastectomy before finally passing away from the disease in 1970. It was a trying time for a teenager, watching my mother deteriorate from a healthy, vital person into a shell of herself. She was robbed of a long life and seeing her children grow into adults; I lost the closest friend I have ever known. But her death allowed my father to marry once again, and I grew to love my stepmother as deeply as my own mother. I was lucky enough to have two mothers who cared for me, and I think of them both often, wondering how my life would be different if I had not experienced it with both of these wonderful women.
Assuming that you are talking about major events in society, rather than individual events (like my father not meeting my mother in December of 1960), then the event would be the passage of the Civil Rights Act of 1964. As a person of color (partly, at least), my life would have been very different if this law had never, to this day, been passed.
Without the Civil Rights Act of 1964, it would still be legal for people to discriminate against me based on the color of my skin. It would be legal for a restaurant to refuse to serve me. It would be legal for the local swimming pool to refuse to allow me to swim there. It would even be legal for a business to refuse to hire me based on my skin color. If my children were deemed to be non-white (they are 3/4 white), they would be liable to the same kind of discrimination.
If we could somehow imagine that the Civil Rights Movement had not worked and the Civil Rights Act of 1964 had not been passed, my life could have been very different.
Woodstock. The "hippy" movement, although extremely rebellious against society, broke through many rules of society back then (males grew their hair long, they wore neon shirts, etc. Not saying I agree with the whole lust part though...) and created the freedom that people today exercise in thinking for themselves. Individuality today (for me)! (Although, I must say that the internet has ruined that individuality a bit...).
All the other posts are probably better than this one though, so don't listen to me.