A split appeared in the Democratic Party in 1948, where southern party loyalists who had voted for Democrats since the Civil War and before were angered by Truman's decision to integrate the military. This led to a third party candidacy by Strom Thurmond on a segregationist platform. This was the beginning of the end of Democratic Party dominance in southern states. By the time Kennedy was elected as a northeastern, pro-civil rights Catholic, the split grew deeper. There was another third party challenge by George Wallace in 1968 that drew considerable support from Democrat Hubert Humphrey, in part allowing Nixon to win.
The long range consequence of the movement of minority support away from Lincoln's Republican Party and towards the Democrat has been the loss of almost all southern states in Presidential elections and most southern seats in the Senate and House to white males.
The Democrats were already starting to build a platform on social change with FDR and the New Deal, but the Civil Rights movement allowed the Democratic Party to really change its image. The party after the Civil Rights movement was a voice for the people in America that do not really have any power or influence. The Democrats showed that they wanted to help middle America and protect the rights for all individuals. The gains from this platform was candidates that really helped to make change in America all the way up to Barack Obama today. The losses are that the Democrats say a lot but do not necessarily walk the walk. They tend to let themselves get bullied by lobbyists and the Republican party so they usually back down. What is different with our current president is that he is not willing to back down and is trying to take advantage of the Democrat majority in Congress. Overall, Democrats have the image of social justice, but they need to actually show they are willing to fight for it!
The Democratic Party's association with the civil rights movement is what has made it into the party that it is today. It is, in large part, why the Democratic Party is the party of non-whites today and the Republicans always win the white (especially the white male) vote.
After the civil rights movement, the Republican Party adopted their "Southern Strategy." In this strategy, the GOP appealed to whites who were turned off by the gains of the black movement and by the lawlessness of the Vietnam Era.
As the GOP took these voters away from the Democratic Party, the Democrats had to become more liberal and more pro-minority. The GOP turned the opposite way.
So the Democrats won the liberal and non-white vote, but they have pretty much lost the conservative and populist white vote for all of my lifetime up to this point.