The portion of the Constitution you are referring to is known as "The Supremacy Clause", and it basically states that the Constitution is the Supreme Law of the Land, followed in order of authority by Federal law, State law, and then finally, Local law. Each level is subordinate to, meaning it must not go against the level of law above it.
This means that a state law that goes against a federal law is unconstitutional. For example, 15 states have laws making medical marijuana legal. At the federal level, all marijuana is illegal, for any reason. When this issue makes it to the federal courts, the state laws will be ruled unconstitutional because of the Supremacy Clause.
Local laws have the least authority, and cannot contradict state law, federal law or the Constitution.
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
The 10th amendment also specifies what kinds of laws are state powers only, and this does protect the authority of the individual states somewhat.