Traditionally, all rights not provided to the Federal Government are given to the states. Typically the kinds of issues states regulate have to do with the welfare of their residents, whereas the Federal Government is not supposed to legislate on the general welfare of the states. The way the federal government typically got around this in the past was due to the Commerce Clause, whereas now the 14th Amendment is commonly used as well to allow the federal government to impose certain laws on the states that used to be solely in the realm of state law.
The struggle between the power of the Federal government and the power of the States has been hotly debated even before the Constitution was drawn up. Back in 1787, America had just won its independence from England, and the war had been started because the States wanted to be free of the strong, heavy hand of the King of England. The Founding Fathers were very wary of creating another strong central government, which they feared could become as oppressive as the British government had been. On the other hand, without some kind of central power, the States couldn't function as a unified entity. So the writers of the Constitution gave the Federal government certain powers, for example coining money, which it took away from the States. But the Tenth Amendment reserved for the States any power which was not specifically granted to the Federal Government.
Between 1787 and 1865, some of the States struggled mightily against Federal supremacy. This led to the "Nullification Crisis" of 1828, when South Carolina passed a law nullifying a tax in imports imposed by Congress. This crisis ended in a compromise between South Carolina and Congress, but the simmering struggle of the States, especially in the South, to chart their own destinies, was the main cause of the Civil War.
Until 1865, the Tenth Amendment was the basis for the argument that the Federal government had no power to overrule a State law. But with the enactment of the 13th, 14th and 15th Amendments, the Federal government was given the power to override State laws that conflicted with the Constitution.
Until 1913, people paid income taxes only to their States, for the most part, and the Federal government earned its income mostly through duties on imports. But with the Federal Income Tax becoming law in 1913, people paid taxes directly to the Federal government, thus greatly increasing Federal power. In addition, our foreign wars, beginning with the Spanish-American war in 1898 and then especially World Wars I and II (and the tremendous expansion of the reach of the Federal government with Franklin Roosevelt's New Deal), developed the Federal government into a world power and forever laid to the rest the concept that America was just a loose organization of sovereign states.
Today, the States retain exclusive control only over a few local issues. Any area in which Congress has announced Federal supremacy is controlled by the Federal government. Therefore, for example, the States cannot pass laws affecting the rights of passengers to sue airlines for injuries, because Congress has assumed control over interstate travel.
The 10th Amendment basically vests all rights not given to the federal government to the states. The 14th Amendment, and specifically the interpretation of the interstate commerce clause has been read by the courts to mean that every citizen of every state is entitled to the same rights, and no state can restrict the rights a citizen enjoys in another state. IN some cases, individual states can confer rights on people which exceed rights provided by other states. For example in Florida there is a written constitutional right to privacy, which does not exist in other states. The issue is a tricky one, and it is probably the hottest issue trickling through the Federal Courts today.
The present U.S. Supreme court has been either schizophrenic or intellectualy dishonest about this issue. In cases where the states have provided caps on tort liability, curtailed certain rights to sue, or allowed big businesses to do things otherwise forbidden, they hold that the State Government trumps the Federal Government. However in the medical marijuana cases, and in Bush v. Gore, where the state clearly had the right to run its own government the Supreme court stepped in and held the states had no such right. I fear this issue is becoming very politicized, and we are losing sight of the law.