In the United States in the 1960s, there were four presidents.
Dwight D. Eisenhower, a Republican was president until January of 1961.
After that John F. Kennedy, a Democrat was president until he was assassinated in November of 1963. He was succeeded by his vice president, Lyndon Johnson. Johnson won his own term in 1964. He chose not to run again in 1968 because of how badly the Vietnam War was going.
The president for the rest of the decade was Richard Nixon, another Republican.
During most of this time, Congress was controlled by the Democrats.
I hope that is helpful... your question is not very clear as to what you want...
In some respects, one could argue that in the mid to late 1960s, the government institutions were more influenced by social movements than anything else. In determining "who" was the government in this time period, a credible case can be made that "the people" composed much of how government functioned in the time period. The growing dissatisfaction with how Black Americans were being treated in America and then how the Vietnam War was being conducted helped to mobilize people from all narratives in demanding change from their government. The leaders of the time were held in check to a great extent with trying to make sense of the intense level of popular and public demonstration and support of what was happening. It ended up determining that Johnson could not run for a second term and dogged Nixon throughout his tenure as President. While there were individuals who occupied the roles of government, a very strong case can be made that the voice of dissent and "the people" were the government in the 1960s.
There were a lot of things going on with the government in the 1960's.
There was a great deal of tension between the Soviet Union and the United States. There was the Cuban Missile Crisis that occurred in 1962 and war was seen as inevitable. Both countries were arming themselves for war but ended up realizing that they needed better relations.
There were also horrible assassinations that took place. President Robert F. Kennedy was assassinated in 1963 and civil rights leader Martin Luther King, Jr. was assassinated in 1968.
Then there was Vietnam. In 1965, President Johnson started bombing North Vietnam. Troops were also sent later that year. Some supported the war and many opposed the war. There were countless demonstrations and protests.