There are a tremendous number of things that happened in the 1960s that were important political events that changed the world. There is no way to list them all in this space. Let me mention a few that will include domestic events in the United States, foreign events involving the US, and events that are not centered around the US.
One of the biggest political events of the 1960s in terms of American domestic politics was the Civil Rights Movement, which culminated with the Civil Rights Act of 1964 and the Voting Rights Act of 1965. This event changed the world because it led to the creation of a new America, one in which all people had legal equality regardless of race. Our country still has many problems that have to do with race, but, thanks to the Civil Rights Movement, there is much more racial justice today than there was before the movement occurred.
In American foreign policy, one of the biggest events was the escalation of the war in Vietnam. If we want to cite a specific event, we can mention the Gulf of Tonkin Resolution of 1964. This allowed President Johnson to take any actions he felt necessary to fight communism in South Vietnam. It changed the world by bringing America into the war on a very intensive level. The Vietnam War changed the way that the US conducted foreign policy for quite some time since the US loss in that war made American leaders somewhat leery of intervening in foreign countries for a while.
Outside the United States, a major event was the building of the Berlin Wall in 1961. By building the wall, the communists created a major symbol of the Cold War. They also changed the world by making it very hard for people to escape from the Communist Bloc. This made the lives of millions of Eastern Europeans more difficult.
All of these were important political events in the 1960s, but this is in no way an exhaustive list of all of the most important events of that decade.