As I mentioned earlier, the Democratic Party had a rule at its first hundred years of national conventions, that a candidate needed to secure a two-thirds vote of the delegates to be nominated for president.
Here's how TIME magazine described the history of that rule, on the eve of its being abolished (the rule, not the magazine), in 1936.
Whatever motives anyone might have had for introducing or perpetuating such a rule, it's clear that one effect of it was to give southern Democrats veto power over the selection of presidential nominees. The allocation of delegate seats to each state is based on a combination of population and of support for the party within that state. During the years when southern Democrats worked to uphold their system of legal racial segregation, the Democratic party won elections in that region with overwhelming majorities. Therefore, those states were overrepresented at Democratic conventions. So, while as a region they had less than 50%, they could easily muster a 33.4% vote to block any nominee unsympathetic to their cause.
Segregationists used a similar tool in the U.S. Senate to block anti-lynching legislation and other civil rights measures. That tool was the filibuster. Unlike the U.S. House, where debate is always subject to strict time limits, the general rule in the U.S. Senate is that any senator can speak for as long as he or she wants. To filibuster is to prolong debate so as to prevent the Senate from taking a vote on passage of a bill. Originally, a minority of one could, in theory, block legislation. Then, in 1917, the so-called "cloture rule" was adopted, allowing the Senate, by a two-thirds vote, to end a filibuster. That requirement was later lowered to three-fifths. But still, a minority of 41 of the 100 senators can hold up Senate action on legislation. Finally, by the 1960s, there was sufficient support for civil rights legislation that the Senate was able to invoke cloture.
On the other hand, the segregationists were quite happy to reinforce majority rule within individual southern states, where they were confident of a majority, especially during the period when they employed several mechanisms to prevent African Americans from voting. I refer to the primary runoff rule.
The Democratic primary at that time and place was, in effect, the general election. The Republican Party was all but non-existent in the south, so the winner of the Democratic primary would face no effective opposition. The general rule in southern states was, and continues to be, that no one can be nominated in a primary with less than an overall majority of the votes. If no candidate exceeds 50% in the primary, a runoff primary is held between the two top vote-getters.
Again, there's no absolute certainty as to the motives of those who established these rules. But here's one academic study that considers many possibilities, while coming to the conclusion that the primary runoff rule was intended to suppress African Americans' political rights.
Consider the example of a primary in which only one candidate is relatively liberal on racial issues, and he is running against four segregationists. The liberal may have received only, say, 25% of the vote. But if the remaining 75% is split among the other four candidates, perhaps 20%, 20%, 20% and 15%, then the liberal candidate could win. But, in reality, the liberal candidate would need to compete in a runoff against the second-place finisher, and he would likely be defeated 75% to 25%.