Describe the two sides of conservatism that contributed to the political thinking of the 1960s and '70s.

The mainstream Republican Party of the 1960s and 1970s would be considered moderate, even somewhat liberal by today's standards. Though one could argue that the split between the "conservative" and "liberal" wings of the Republican Party started after the Second World War, for brevity, we will stick to the era in question.
New York Governor and Vice President Nelson Rockefeller, California Governor Ronald Reagan, and former Vice President and future President Richard Nixon were alike in terms of their strong authoritarianism, which was appreciated by conservative white voters who were alarmed by uprisings in urban areas and unrest on college campuses.
On the other hand, the mainstream Republican Party was not always averse to social programs or even the protection of rights for groups toward which they might otherwise have been averse. For example, Nixon created the Environmental Protection Agency (EPA) on December 2, 1970. During his years as governor of California, Reagan helped to defeat the Briggs Initiative, a ballot proposition sponsored by conservative Orange County state legislator John Briggs that sought to ban gays and lesbians from working in public schools. Reagan formed an unlikely alliance with President Jimmy Carter and fought the initiative on the grounds that it violated free speech, given that those targeted by the ballot measure would have to inhibit themselves from saying anything about their orientations. Another rights argument was that it violated equal protection under the Fourteenth Amendment.
In New York, Rockefeller expanded the state university system. While running for the Republican nomination in 1964, he supported civil rights as well as initiatives for cleaner air and water. His opponent, Barry Goldwater, reflected the more conservative—or extreme—strain that existed within the Republican Party.
Goldwater's politics were fervently anti-Communist. He strongly opposed the social programs proposed by Democrats, whom he believed were trying to turn the United States into a socialist state. He also opposed the passage of the Civil Rights Act of 1964, believing it to be an example of the federal government's overreach into states's rights.
Goldwater's views, arguably, provided a blueprint for many members of today's loosely constructed Libertarian Party, which, like Goldwater, strongly supports gun rights and is very watchful of government interference in private and local affairs. In terms of civil rights, both Ron Paul and his son, Rand Paul, have voiced opposition to the "federal overreach" of 1960s civil rights legislation.
To summarize the two sides, one side (Rockefeller, et al.) was composed of Republicans who believed in a degree of social engineering that reflected conservative values; the other side believed that the government should not interfere with states's rights, even when it seemed to oppose practices in those states. Therefore, Goldwater probably would have opposed the interference of President Carter in the possible passage of the Briggs Amendment, thinking that such decisions should be left up to the people of the state.