Summarize the Brown v Board of Education case. Based on the case outcome, explain why the Little Rock Nine situation still happened. and what a movement is.

Brown v. Board of Education was a landmark case in United States history as it arguably launched the civil rights movement. The Little Rock Nine were the first black students to enact this ruling of the Supreme Court.
Brown v. Board of Education was a consolidation of cases from five jurisdictions that concerned the same problem of segregated schools:


Brown v. Board of Education (Kansas)

Briggs v. Elliot (South Carolina)

Bulah v. Gebhart and Belton v. Gebhart (Delaware)

Davis v. County School Board of Prince Edward County (Virginia)

Bolling v. Sharpe (District of Columbia)

At the time of this case, most of the segregated schools were not "separate but equal." Often the buildings of the black schools were in ill repair, books were the ones used for years by the white schools and were dirty and torn, and indoor toilet facilities did not exist. Nonetheless, the purpose of the consolidation of these cases was not the physical conditions, but the fact that segregation itself was "inherently unequal and a violation of the equal protection clause of the 14th amendment of the Constitution."
The unanimous decision of the Supreme Court in Brown v. the Board of Education was that an education of quality was essential for all children. Therefore, the court ruled that the states must ensure this educational equality for everyone regardless of color or creed. Chief Justice Earl Warren wrote that all children must be provided an adequate education regardless of their color if they are to succeed in life:

Such an opportunity, where the state has undertaken to provide it, is a right that must be made available on equal terms.(1954)

The unanimous decision of the court in 1954 was, in essence, that "The doctrine of separate but equal has no place" in education.  
Thus, because the resistance to this ruling was so widespread in the South, there was a second decision made in 1955, known as Brown II. This decision ordered school districts to integrate "with all deliberate speed." As a result of this decision and because there was pressure placed upon the school board of Little Rock, Arkansas, by the NAACP, preparations were made to enroll African-American students at the high school in order to put the new law into effect. Among those interviewed, nine students seemed stalwart enough to withstand the attacks to which they would be subjected, and they enrolled at Central High School in Little Rock, Arkansas.
Very soon thereafter, two pro-segregation groups formed to oppose the plan of the NAACP: the Capital Citizens Council and the Mother’s League of Central High School. So, the nine students were counseled on how to respond to hostile attacks. Then, on September 2, 1957, Governor Orval Faubus of Arkansas promised to call in the Arkansas National Guard to prevent the African-American students’ entry to Central High. But, Judge Davies, a federal judge, issued an order that the nine students would, in fact, enter the school. However, despite this order, the National Guard prevented the students from entering that day.
During the following weeks, Judge Davies commenced legal proceedings against Governor Faubus. President Eisenhower then tried to persuade the governor to remove the National Guard, but federal Judge Davies finally had to order the Guard removed. When this occurred, the Arkansas police had to escort the students into the school through an enraged mob. Finally, the students had to be escorted away. The following day, the students were again escorted by federal troops into the school. While they remained at Central High School throughout the year, the National Guard was present every day at this school for the rest of the year. Still, the nine students were met with hatred and acts that harmed some of them throughout the year, but they remained. They, then, were responsible for the movement, or change in social ideas, that began to take place throughout the South.
After the school year was finished, one of the nine students graduated. Then, in September of 1958, Governor Faubus closed the Little Rock high schools, making it necessary for the remaining eight students to receive their education in other schools or through correspondence courses.  
https://www.history.com/topics/black-history/central-high-school-integration