The 1950s and 60s were part of a party realignment process that began in the 1930s and ended around the early 70s. This process involved shifts in both elite-level coalitions and mass public opinion from Progressive Era alignments--a solid Democratic South, industrial and business-oriented Republicans popular in the Northeast--towards what we might recognize today--a Democratic Party including African Americans and racial liberals, labor unions, and urban voters, and a Republican Party growing stronger in the South with an emphasis on conservatism (racial and otherwise) and states' rights.

While there were signs of these new alignments as early as the 1930s, this process occurred unevenly across states. Consolidation of coalitions and issue positions on a national level only began to become clear in the 1970s. What it meant to be a Democrat or Republican varied widely by geography during this period. For example, the New York State Democratic Party was allied with civil rights and labor groups starting in the 1940s; in the South, the Democratic Party had been so dominant for so long that Southern segregationists remained in the party until or even after Democrats became the party of racial liberalism at a national level. Clearly, being a Democrat in 1960 would have meant something very different to a voter in New York than to one in Alabama.

This national variation led to some confusion, or at least inconsistency, among the mass public about what it meant to be a Democrat or a Republican. The reasons for party identification given by contemporary scholars in the 60s are similar to the accounts we have now: social characteristics of individuals and their surroundings cause people to form attachments to parties that are generally stable over time. However, since the relative meanings of the parties were more varied and less clear than they are now, it's harder to generalize about what associations people formed with parties then.

For a thorough and digestible treatment of the realignment process, see Eric Schickler's Racial Realignment: The Transformation of American Liberalism, 1932-1965. For a hugely influential account of voters in this period, see Campbell, Converse, Miller, and Stokes's The American Voter.

There are many reasons, some of which might vary depending on where you live.

The South, for example, was still solidly Democratic for much if this period, largely because it was the party that defended segregation: the Republicans were the party of Lincoln, after all, and Southern Democrats were extremely protective of state's rights and segregation itself. What's more, these states tended to be ruled by powerful Democratic "machines," whereby the party controlled a huge number of patronage appointments/positions and used that to control the state.

As you can imagine, this caused huge tension within the party as Northern Democrats became increasingly pro-civil rights beginning in the 1950s and culminating in LBJs approval of the civil rights act and voting rights act of 1964/5. The Vietnam War gets all the credit for this, but tension between pro-segregation southerners and pro-civil rights northerners was another major reason why the 1968 democratic convention was such a shit show. There were periodic insurgent or third-party pro-segregation candidates like Strom Thurmond and George Wallace who sometimes threatened the "solid south" for the Democrats, but in general this did not really begin to change until the 70s, when southern states first started to go to republican presidential candidates (ie, Nixon's in 72, then Reagan in 1980).

So that's one answer, which applies to a specific region. But the larger answer is pretty straight forward: the democrats were the party of liberalism, and had been since FDR. The single biggest, most important difference between Republicans and Democrats during the 1950s and 1960s was that Democrats believed in an activist government: one that would actively intervene in society to uplift the poor, to protect the rights of the vulnerable, and to take care of everyone's most basic needs. Basically, they believed that the government should use regulation, law, and government funds to create a fair, more equitable society free from want and exploitation. They were (mostly) pro-union. You see this ideology expressed in Democratic policies thought the period: the New Deal, the "Great Society" reforms, the welfare state in general. Republicans, in contrast were pro-business: they were wary of ambitious government programs aimed at changing society or eliminating want, believing that the free market and private enterprise would uplift everyone.

Between FDR and either the election of Nixon in 1968 or the election of Reagan in 1980 (depending on how you slice it) Democrats presided over what's known as the "New Deal Coalition:" this was made up of white working class people who saw unions and the government as their best defense against exploitation by businesses and corporations (a ''huge'' constituency at this time, much larger than it is now), civil rights supporters, immigrants, liberal intellectuals, and southern Democrats. The party built a very "big tent" during this period that allowed them to appeal to a wide variety of constituencies, most of whom (with the possible exception of Southern Democrats, who loved public works and such but ardently defended states rights) believed in an active, interventionist government.

There's been a lot of fascinating scholarship trying to explain why the New Deal coalition collapsed lately, most of it focusing on why the white working class deserted the democrats. There a lot of possible explanations there, but in general historians have focused on resentment over bussing (forced desegregation of suburban schools) and welfare programs (ie, "why does the government cut checks for poor, urban blacks who don't work while we work hard, but are still struggling, and get nothing"), and a kind of latent social conservatism/anti-intellectualism among working class whites that the democratic leadership underestimated, but which republicans like Reagan and Nixon realized they could harness. Another explanation has to do with economic transformation and regional migrations: the de-industrialization of the north hollowed out the comfortable middle- and working-class white constituencies that had been a big portion of the democrat's support in the north-east and mid-west, while at the same time the rise of "sunbelt" cities in the south and south-west fostered new, socially conservative, pro-business constituencies in the south.

The "rise and fall" of the new deal coalition has been a huge topic of study by historians over the last ten years, see Jefferson Cowie's books Staying Alive and The Great Exception or him and Nick Salvatore's "The Long Exception" for a more thorough account of this. There's also a good (textbook) overview called The Rise and Fall of the New Deal Order but it's kind of old now, and there's a lot of exciting newer scholarship like Robert Self's All in the Family that complicates the picture too.

Edit: Forgot to mention one very major reason why people would vote Republican during this period: they had a very large, very vocal wing of the party whose primary concern was anti-communism and foreign policy. So they pulled in huge constituencies of Cold War hawks and people who were ardently opposed to communism and/or socialism.

To attempt to put it clearly - it was basically minimal-regulation free marketeers (or 'pro-industry') on the right vs. the working class vote on the left (pro-welfare, etc.).

Later on, the left jumped on the civil rights band wagon. However, the working class south wasn't for that (traditionally speaking), so the right started to represent them by creating socially conservative platforms.

Bit of an awkward shift, which might explain why american politics either feels really dumb, or overly mixed up in terms of party and ideologies.