Both parties flipped their views on segregation, during a period when it was ending throughout the country. This was the key issue in the South. For the rest of the country, nuclear weapons were a key issue, but the country's view of them changed during those 8 years.

Integration and the Civil Rights Movement were crucial to this period of American history. The end of segregation and Jim Crow laws was extremely unpopular among white Southerners, and they supported candidates who would be likely to resist it. Eisenhower and Stevenson both acted moderately on the issue, but Eisenhower was generally in favour of integration and Stevenson urged caution and a very slow end to segregation. Southerners supported the Democrat, Stevenson, but Eisenhower was overwhelmingly popular with the rest of the country and won easily. However, in the 1964 election, the Democrat Johnson was the President who had passed the Civil Rights Act. Goldwater had voted against it, arguing that states had the Constitutional right to choose their own stance on segregation. White Southerners who considered segregation the most important issue supported Goldwater, while the rest of the country voted for Johnson.

Nuclear weapons seem less important now, but were then a focal point of both elections. Eisenhower threatened to use nuclear weapons during the Korean War, which led to an armistice that ended the conflict. Americans considered The Bomb to be a useful threat to wield against other nations, and they wanted a Republican president who would threaten to use it. But in 1961, the Soviet Union became the first country to put a man into orbit. Also in 1961, the Soviets tested a 57-megaton nuclear bomb, the most powerful that the world had ever seen. A rocket that could put a man into low-Earth orbit could also land a hydrogen bomb on the United States, and average Americans no longer felt impervious behind their nuclear arsenal. Goldwater wanted to overpower the Soviet arsenal by building more American nuclear weapons, and he stated that he would be willing to use any weapons necessary to win the war in Vietnam. The Johnson campaign painted him as an unstable warmonger who might cause a nuclear apocalypse. Americans no longer felt that they could win a nuclear war against the Soviets, and they voted for the Democrat who promised a more diplomatic approach.

More of course can be said, but our FAQ has some older entries which can at least start you off.