For this answer, I will assume that you are talking about politics in the United States.
The obvious place to start with this is to look at the three men who won presidential elections in the 1960s. These were John F. Kennedy in 1960, Lyndon B. Johnson, who became president when Kennedy was killed and who then won the 1964 election, and Richard Nixon, who lost to Kennedy in 1960 and won the presidency in 1968. These men were all extremely influential in the politics of the 1960s.
After those three, it is a little harder to know who to pick. One good person to select would be Martin Luther King, Jr. King was, of course, the major figure in the Civil Rights Movement. That movement was important for the first half of the 1960s, culminating in the passage of the Civil Rights Act of 1964 and the Voting Rights Act of 1965. In addition, King deeply influenced politics indirectly when he was assassinated in 1968. His assassination set off riots in cities around the country, helping to deepen divides between blacks and whites.
There are many other possible choices. I would go a little outside the box for my last choice. I would choose General William Westmoreland. He was the general in charge of American forces in Vietnam. His conduct of the war and his frequent requests for more troops to be sent to Vietnam strongly influenced politics in the second half of the 1960s.
Thank you so much!