The Civil Rights Act of 1964 is a great example of federalism in action. A basic tenant of federalism is that federal laws supersede those of the states and municipalities. The main focus of the Civil Rights Law of 1964 was to protect the rights of minorities that were being denied at the state and local levels. Pursuant to the Fourteenth Amendment, which guarantees equal protection under the law, this act sought to overturn segregationist Jim Crow laws that were prevalent in much of the South.
After the Supreme Court overturned Plessy v. Ferguson with Brown v. Board of Education in 1954, it became incumbent upon the federal government to ensure that that equal protection was not denied to Americans as a result of segregation. Although some local laws already existed to provide protection against certain forms of discrimination, it was generally agreed that these were not comprehensive enough to be meaningful. Jim Crow policies and laws still allowed for and promoted segregation throughout much of the country. Therefore, civil rights advocates pushed for a federal set of laws that would take precedent over those at the state and local levels.
Almost immediately after the passage of the Civil Rights Act, there were state and local challenges to the new laws. They often claimed that the Civil Rights Act was in violation of the Tenth Amendment which should leave such matters to the states. While state courts sometimes upheld these challenges, when they made it to federal courts, segregationist policies were struck down. Chief Justice of the Supreme Court Earl Warren was particularly keen to apply a federalist view of this law. In the case of Heart of Atlanta Motel, Inc v. United States, the Supreme Court voted unanimously in support of the federal government's power to enforce desegregation of many public accommodations pursuant to the Commerce Clause of the Constitution.
Given all of this, the Civil Rights Act of 1964 illustrates the federalist notion that state and local laws must conform to federal laws as supported by the Constitution and that federal courts have the power to enforce this.