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.