When American conservatives talk about states' rights what are they referring to and how does it relate to the Constitution?

Answer 1

There are two questions here:

What are the Right-Wing Candidates talking about when they refer to States' RIghts?
As concerns what the US Republican Candidates are referring to, that is anyone's guess. They are usually calling "State's Rights" on questions that have relevant Federal Laws that already apply to them or are based on questions that never applied to individual states. At best, it could be some sort of plea for a reinstatement of the consistently derided Nullification Doctrine, which is the idea that states can violate federal laws if the state government is not pleased by them. This has consistently been refuted by nearly every court that has heard the argument, primarily because of the Supremacy Clause of the Constitution, which holds that on matters about which a federal law can be made, federal laws are superior to state laws in every conflict.

What are States' Rights?
Regardless of the failure of most Republican Candidates to properly articulate any legal concept in the United States, States' Rights are an actual political concept in the United States. Simply put, States' Rights refer to the ability to make distinct laws in the various states that are different than those applied elsewhere and preserve the particular uniqueness of that state, in contrast to overriding federal law.

In contrast to most countries in the world, which are unitary states, the US is a federation. In a unitary state, like South Korea, the law only really comes from the national government (in Seoul for Korea) and the various provinces or states only exist for managerial purposes. As a result, there is one legal corpus for the entire country. In a federation, like the United States, the national government is a limited government that can only make laws on certain issues, such as foreign relations, currency, and interstate commerce, but cannot make laws on other issues such as marriage, criminal law, or contract law, unless a connection can be drawn from these other issues back to the limited cases provided earlier. For example, while the federal government cannot make laws on marriage as an issue in and of itself, the federal government does have the ability to make sure that laws in the US are not substantively discriminatory, so the decision of the Supreme Court that barring gay marriage is discriminatory is in pursuit of the limited federal interest of removing discrimination, not adjudicating marriage.

The types of arguments that expand federal power into previously illegal territory, such as the argument that the federal government can determine gay marriage to be legal in all states when the federal government does not explicitly have legal permission to interfere with marriage (considering the Tenth Amendment), come at the cost of state power. Since federal laws are superior to state laws on any given issue, any increase in the ambit of things about which the federal government can make laws is a decrease in the ambit over which states have local control. American Conservatives are increasingly finding that the federal government has political aims that run contrary to their local political aims and, therefore, contest these encroachments into what has historically been state-only legal territory. This comes up in questions about marriage, abortion, criminal law and incarceration, education, gun control legislation, and state welfare agencies. These are all cases where the federal government has no historic right to make legislation, but has done so by linking it to areas in which they are permitted to make legislation, such as interstate commerce, due process under the law, the right to privacy, the taxing power, and promotion of the general welfare. The reversal of this trend is what States' Rights is about.