The holding in this case was that segregated public schools violated the 14th Amendment to the Constitution. By holding this, the Supreme Court was overturning its decision from the 1896 case of Plessy v. Ferguson.
In Plessy, the Supreme Court held that segregation (at that point, it was segregation in train cars) did not violate the 14th Amendment. It held that separate facilities for the two races were acceptable so long as they were equal. This was the “separate but equal” doctrine which allowed legalized segregation to occur for so long.
In Brown, the Supreme Court overturned Plessy. It held that, in the case of public education, at least, racially segregated schools were inherently unequal. Since the 14th Amendment requires that all people enjoy the “equal protection of the laws,” this was a violation of the 14th Amendment.
Thus, the Supreme Court held that governments could not run segregated schools. This decision helped bring about desegregation in a number of other areas of public life.