The Supreme Court overturned the "separate but equal" doctrine established by Plessy v. Ferguson in determining that segregation in public schools was illegal, but it did not at first stipulate when schools should be integrated. When it did, a year later, it only said that schools should be integrated "with all deliberate speed." This obviously left much up to interpretation, and local school boards predictably dragged their feet in bringing about integration.
The reason that they did so was that popular opinion in the South was strongly against integration in public schools. White Citizen's Councils were formed throughout the South to resist politically, and many attempts at integration were met with angry, threatening mobs, most famously in Little Rock, Arkansas. In that case, President Dwight Eisenhower sent in federal troops to enforce integration, but this did not happen everywhere. Integration, then, proceeded exceedingly slow in many sections of the South despite the Brown decision.