Answer:
The landmark case that desegregated schools was Brown v. Board of Education, a 1954 case in which the Supreme Court Justices unanimously ruled segregation in the public schools is unconstitutional, and that the African-American students could attend the originally segregated white schools.
Chief Justice Earl Warren, in writing the Court opinion, declared "separate educational facilities are inherently unequal" because they violated the Fourteenth Amendment Equal Protection Clause. This overturned the 1896 Supreme Court ruling in Plessy v. Ferguson, which held the concept of "separate but equal" was constitutional.
While this ended de jure (legal) segregation, the Supreme Court didn't outline the method for desegregation until Brown II (Brown v. Board of Education II, (1955), the following year.
Because the Supreme Court lacks the authority to enforce its rulings, many school districts resisted integration. The federal government didn't enforce Brown I and II until it passed the Civil Rights Act of 1964.
One of the most notable clauses from the majority opinion in Brown II was for the schools to desegregate "With all deliberate speed." Which meant that nothing had to be done immediately or with any sort of urgency, as they did not outline the methods for desegregation as stated above.
Case Citations:
Brown v. Board of Education, 347 US 483 (1954)
Brown v. Board of Education, 349 US 294 (1955)
For more information about Brown v. Board of Education, see Related Links, below.