The discontent of Southern Democrats began with Harry Truman's mild support for Civil Rights. Many Southern Democrats walked out of the 1948 Democratic Convention and formed the Dixiecrat party which ended up carrying Alabama, Louisiana, Mississippi and South Carolina that year. In 1952 and 1956, Eisenhower was able to break into the South winning a few states.

But it was Kennedy and then Johnson's support for Civil Rights that really irked Southern Democrats. But Republicans were still the party of the elites, so once again they formed their own party, the American Independent Party, headed by George Wallace the Governor of Alabama and a staunch critic of Civil Rights.

But, Nixon realized this amazing opportunity to pick up tons of electoral votes simply by pushing law and order (it's worth noting here that the Civil Rights act was followed by massive upheaval in the United States with devastating riots in LA, Chicago, New York, etc.). So, in 1968 the South was split between Nixon and Wallace, with the Democrat, Hubert Humphrey losing every Southern state.

That was the end of Democrat dominance in the South.

It's important to note, however, that it wasn't just race! That was definitely a driving cause but there were a few other things: The south during the 20th century saw an influx of migrants from northern states who carried their Republican beliefs with them. On top of that, income in the south improved, so they were no longer as firm believers in the sort of big-government ideas of the Democratic Party. And, it helped that the Democrats shifted to the left on things like abortion. The South has always been deeply religious and that didn't sit well with them.

If you want to learn more about that era, find a copy of Nixonland by Rick Perlstein -- it's so good. (It's worth mentioning that he's pretty liberal, but I don't think that's evident in the book).