Division of Powers Between State and Federal Governments
There are some powers that are reserved solely to the federal government, some that can only be exercised locally or by the states, and yet others which can be (and are) exercised by all of the above.
Powers Reserved Solely to the National (Federal) Government
Several powers are reserved by the U.S. Constitution exclusively to the national government (commonly called the 'federal' government.)
- Foreign relations, especially treaties
- Coining money
- Imposing duties on imports or exports
- Keeping troops or ships of war during peace time
- Engaging in war, unless actually invaded or in imminent danger
These limitations on state governments are laid out in Article I, Section 10 of the Constitution of the United States.
Powers Reserved to the State and Local Governments
The Tenth Amendment to the Constitution of the United States (the last of the ten amendments known as the Bill of Rights) says:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
So, in theory, any power that isn't explicitly granted by the U.S. Constitution to the federal government is reserved to the states and not available to the federal government. The exception is when a power is not granted to the federal government and is explicitly banned by the Constitution to the state governments, in which case neither may exercise that power. The latter would include cases such as granting titles of nobility or making ex post facto laws, for example, which is not possible for any level of government in the U.S.
In practice, however, this is more murky, as courts have occasionally allowed the federal government to exercise powers in all sorts of different things under incredibly broad interpretations of the Interstate Commerce Clause. This is why all sorts of federal laws and regulations will contain language similar to "when engaging in interstate commerce" or "for the purposes of engaging in interstate commerce," since the federal government technically has no Constitutional authority to regulate commerce within a state.
For example, even federal drug laws are deemed to be authorized under the Commerce Clause, as they regulate commerce of the drugs. Courts have further ruled in many cases that trade within a state may be regulated where it's deemed necessary to regulate interstate trade.
Powers Exercised at Both State and Federal Levels
There are many areas where federal powers overlap state and/or local powers. Drug laws are again an example of this, as are many criminal laws. So long as the two do not disagree, both are applicable and cases could be prosecuted under either. The lack of a federal law does not ban a state (or even a city or county) from making a law, though the federal government could explicitly make a law banning states from making a law restricting something.
For a more in-depth look at which government services are run and funded by which levels of government, see this answer on skeptics.SE.
What if State and Federal Law Conflict?
Article VI of the U.S. Constitution states that:
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, anything in the Constitution or laws of any State to the contrary notwithstanding.
So, if state and federal law disagree on a matter where the federal government has power to make that law, the federal law wins.
Again, however, what happens in practice can be different. In the particular example of laws regarding marijuana, there is a federal law banning its possession or sale in most cases, but some states do not impose a state law against it. In these cases, state and local law enforcement normally would not make arrests on these crimes. Federal law enforcement still could, however, under the federal laws. But the executive branch of the federal government could decide (and, in this case, has decided) to not do this. Thus, it's still technically illegal in the entire U.S., but those laws are effectively not enforced in the states that have no law against it.
The federal government could, however, decide to resume enforcing these laws any time it wanted to and the states would have little-to-no recourse to stop it from doing so, other than attempting to argue in court that federal drug laws aren't authorized under the Constitution in the first place. This would be a difficult case for the state to win in the current legal environment of broad interpretations of the Commerce Clause.
Additionally, if a state law and a local law conflict, it is also usually the case that the state law will preempt the local one. Unlike the federal government, state governments are not constrained to the powers enumerated in the federal constitution. In general, the only way for a local law which contradicts a state law to stand in a challenge would be for the state law to be struck down, such as if it violated some provision of the federal constitution or of that state's constitution. Otherwise, the state law will preempt the local law.
Who decides if they really conflict?
If a state law or local law is thought to be in violation of a federal law or the federal Constitution, it can be challenged in court as such by someone who is deemed by the court to have standing to bring the case. The courts will then rule whether or not the state law violates the federal law and, if it does, the courts will strike down the state law. If the parties still disagree, they can appeal the decision to higher appeals courts, possibly all the way up to the U.S. Supreme Court. The Supreme Court has the final say in this, as described in Article III of the U.S. Constitution.
On the flip side, if the federal government creates a law that is thought to exceed its enumerated powers, someone with standing can challenge it in court. If the courts rule the federal law exceeds the power of the federal government, the federal law will be struck down. Again, one or more states could potentially themselves be the party with standing to bring the suit here. And again, rulings can be appealed through appellate courts, potentially up to the U.S. Supreme Court, which would have the final say as per Article III.
Once the U.S. Supreme Court has ruled on a matter of U.S. Constitutional law, the only ways to change it are either for the Supreme Court to overturn their decision in a later ruling on a different case or for the U.S. Constitution to be amended, according to the process set out in Article V.
What if the State or Federal Law is More Restrictive than the Other?
If a state law is more restrictive than a federal law, but does not contradict the federal law, then the more restrictive state law applies (and the same goes for local laws.)
In the particular mentioned example of federal vs. state (vs. local) minimum wages, the most restrictive (highest) minimum wage in a given jurisdiction would apply. So, if the federal government has a $7.25/hr minimum wage, but the state government has as $10 minimum wage, the minimum legal wage in that state is $10. If a city in that state then decides to impose a $12 minimum wage, then the minimum wage inside that city is $12.
Is the Division of Powers Settled or Controversial?
Some parts are very unambiguous and settled (for example, no states are going to be coining money or declaring war any time soon,) but others remain quite controversial.
In general, those who follow an originalist philosophy of legal interpretation tend to view the legal role of the federal government in a more limited manner than those who do not. Conservatives and, especially, libertarians tend to favor originalism, while liberals tend to favor loose constructionism. As a result, conservatives and libertarians typically take a more narrow view of the legal powers of the federal government than do liberals.
The Commerce Clause
One of the largest points of contention is, as previously alluded to, the meaning of the Commerce Clause. Loose constructionists (mostly liberals) tend to view the powers granted to the federal government by the commerce clause much more broadly than do originalists (mostly conservatives and libertarians.)
A relatively recent example that highlights the disagreement in regards to the Commerce Clause was National Federation of Independent Business v. Sebelius (Wiki). In this case, the federal mandate for individuals to purchase health insurance or else pay a penalty was challenged in court. The Supreme Court's interpretation of the Commerce Clause here was divided 5-4.
Four justices (from the liberal wing of the court) held that the mandate to purchase insurance counted as a regulation of commerce under the Commerce Clause. However, the majority of the court (the 5 more conservative justices) rejected that interpretation, ruling that the Commerce Clause can't be used to force someone to unwillingly engage in commerce, but rather only to regulate existing commerce. Ultimately, however, a majority of the court upheld the penalty by ruling that it was a tax and, thus, authorized by the taxation clause.
This case also brings up an interesting example of the differences between federal powers and state powers. While, as the court ruled, the federal government has no power to force someone to engage in commerce, due to the lack of an enumerated constitutional power allowing it to do so, this limitation applies only to the federal government. On the other hand, nearly all, if not all, states already have laws requiring the purchase of automobile insurance and these laws are not set up as a tax. So, this is an example of a power that the state governments can wield, but not the federal government.