Political scientists generally identify 5 party systems or eras since the founding of the American republic. The Fifth Party system is generally identified from 1933 to 1968, the time period you asked about. During this time, the Great Depression is going to kick off the major differences between the Democrats and the Republicans. During the 1932 presidential race, President Hoover (Republican) argued that economic conditions would improve on their own and that government intervention would be costly and do little good. Candidate Franklin Roosevelt (Democrat) argued that the government needed to implement programs that would help those unemployed and to spur economic growth. Roosevelt introduced his platform as a “New Deal” for America. After his win in 1932, the New Deal Coalition was credited for Franklin’s win. This coalition included a wide range of interests, including African Americans, Catholics, Jews, union members and white Southerners. These groups became strong supporters of FDR and the Democratic Party in general. This large group of supporters did not always agree with each other, which is why we see some conservative-leaning Democrats in the same party with liberal-leaning Democrats. This coalition is going to hold firm until the 1960s, when race and race relations came to the forefront of American politics. The tipping point scholars often note is the signing of the Civil Rights Act and Voting Rights Act by President Lyndon Johnson (Democrat from Texas). This move by a southern Democrat to sign legislation extending rights to African Americans spurred conservative whites to begin moving from the Democratic Party to the Republican Party. This move would take several decades.

Sources: Kathryn Moore’s The American President; Barbour and Wright’s Keeping the Republic: Power and Citizenship in American Politics