The Supreme Court case of Plessey v. Ferguson basically said that segregation laws found mostly in the south were constitutional. Besides battling segregation in the courts, civil rights leaders such as Martin Luther King, Jr., used non violent protest to try to end segregation. Martin Luther King, Jr., was an educated man and was able to see how successfully peaceful, non violent protest worked with Gandhi in India in their struggle for independence. He knew that in order to succeed in gaining Civil Rights for African Americans he would need the support of the general public within the United States. He knew that if he used violence, the general public would condemn his movement. Martin Luther King’s struggle for Civil Rights gained sympathy and support when the public saw the violent reaction to his non violent protests. These non violent protests led to the support needed to end segregation in the south, overturning Plessey v. Ferguson.
There were a number of steps that were taken to eliminate the ruling from this case. You might want to check your textbook to see exactly how your book talks about these steps.
The basic strategy was to start with lawsuits against "separate but equal" in higher education and then work down until there was a basis for bringing lawsuits to integrate high schools and elementary schools.
So, for example, the NAACP sued over whether it was legal for Texas to maintain two different law schools -- one for blacks and one for whites. The point of these suits was to argue that it was really impossible to have separate schools be truly equal. If separate schools could not be equal, then the whole doctrine of Plessy (saying that separate schools were equal and therefore okay under the Equal Protection Clause) would be destroyed.
This strategy culminated in the Brown case which overturned Plessy.