The verdict of the Plessy v. Ferguson case was that "separate but equal" laws were justified or allowed to exist. This basically meant that states had the right to keep blacks and whites separate at separate facilities as long as they were equal.
Unfortunately, the separate facilities were not equal for blacks and there was terrible treatment and equality for a long time in American history throughout the 20th century.
When the Civil Rights Movement became more popular in the 1950s, decisions like Plessy v. Ferguson were overturned, allowing blacks and whites to go to the same schools. This started with the case of Brown v. Board of Education in 1954, and other schools began to integrate after this important case.
There were still issues with people being treated equally, even though it became the law to treat people equally. With the Civil Rights Act of 1964, this outlawed all segregation in public places, so there would be no more segregation at restaurants, movie theaters, etc.
Some states tried to block this integration with their own ways and laws and in some places, things turned violent with race riots.
Even though places were being integrated at a faster pace, it was difficult for some people to accept the views of everyone being equal.
