The Constitution reflects general principles of federalism -- the idea that multiple levels of government control the same territory. Because of this, U.S. citizens are subject to both state and federal law. The federal government passes laws that address issues of national concern pursuant to powers specifically listed in the Constitution. As sovereign entities, states remain free to regulate matters that don't extend beyond their borders using appropriate legislation.
Article VI of the Constitution, known as the Supremacy Clause, states that the Constitution, federal laws and treaties are supreme law in the U.S. This means that if a state law conflicts with federal law or is unconstitutional, it can't be enforced. Sometimes a federal law states explicitly that it preempts certain types of state laws. Even if a federal law doesn't explicitly preempt a state law, if it is impossible to follow one without breaking the other, the federal law trumps the state law. Additionally, the Constitution requires states and their courts to recognize and respect laws, records and court decisions of other states.
The federal government can enact laws that govern the entire country, but only if they relate to one of the powers specifically granted by Article I of the Constitution. Many of these powers overlap with state powers. For example, Congress can regulate interstate commerce. However, the Constitution does not give Congress power to regulate commercial activities that take place within only a single state. Some federal government powers are exclusive, and the Constitution forbids states from exercising them. For example, only the federal Congress can mint money or declare war.
Unlike the federal government, state governments aren't limited to any specific constitutional grants of power. Rather, the 10th Amendment makes clear that states retain all powers that haven't been granted to the federal government or forbidden to them by the Constitution. While the state's "police power" isn't addressed anywhere in the Constitution's text, it's understood that states have the inherent power to manage matters that take place within their borders. For example, road maintenance and public schools are traditionally considered state issues. Some powers, such as the ability to raise government revenue through taxation, are shared by both state and federal governments.
Complexities of State Sovereignty
Some traditional state functions are considered beyond the power of the federal government to regulate. For example, courts ruled the federal government had no constitutional authority to set minimum pay rates for state government employees. Other state sovereignty issues arise with application of the 11th Amendment, which prohibits individuals from suing other states for damages in federal court. This state sovereign immunity protects states from being sued by the citizens of another state. However, citizens may sue their own state for constitutional violations, such as a violation of their due process rights under the 14th Amendment.