The 1958 Cooper v. Aaron case was one of the most significant Supreme Court decisions of the 20th century. Essentially, the Supreme Court declared that its decisions are legally binding on all states, even those who disagree with its decision.
The Court's decision in Brown v. Board of Education (desegregation of public schools) was unpopular in most of Arkansas. Consequently, the Arkansas state legislature amended the state constitution such that desegregating the schools would violate the state's constitution. In so doing, Arkansas implicitly declared it had the right to nullify federal court decisions it did not agree with.
However, all nine Supreme Court Justices rejected Arkansas' argument. They said that, since the Supreme Court was the ultimate interpreter of the U.S. Constitution, state constitutions must not contradict the Supreme Court's interpretation of the U.S. Constitution.