The conflict between the rights of the states and the powers of the federal government played a major role in the writing of the constitution and the process of ratification. Federalists argued for a strong federal power while Anti-Federalists opposed the new constitution entirely, or at least they wanted a Bill of Rights that would ensure individual liberties. Included in the Bill of Rights that was adopted was the Tenth Amendment, reserving all unallocated rights to the states and the people.
Some of the anti-federalists had been prominent in the American Revolution, and didn't feel that the Articles of Confederation needed to be changed at all. Patrick Henry was one of the prominent members of this group.
In the decade immediately following the ratification of the constitution, one of the issues that divided opinion was the passage of the Alien and Sedition Acts. Viewed by federalists as a means of protecting the national wellbeing, it was regarded by Jefferson and Madison as an attack on American liberties. Between them, they wrote the Virginia and Kentucky Resolutions, which stated not just their opposition to those acts but to the encroachment of federal activity into state domains.
The Virginia and Kentucky resolutions put forward a theory of Nullification, which Jefferson made even more pointed in his 1799 Kentucky resolution. The issue that was most important through the Andrew Jackson era was tariffs, which were resisted in the South and which brought about the nullification crisis.
In 1832, John C. Calhoun resigned the office of vice-president in reaction to the Tariff of 1832. With his encouragement, his state of South Carolina passed an ordinance of nullification. Calhoun wrote an Address to the People of the United States in which he explained the objections to the tariff and what he held to be the rights of states. The latter became a favorite theme of states' rights advocates.
The declaration by South Carolina was regarded by President Jackson as a threat to the national security and unity. He responded with his own Proclamation to the People of South Carolina, on December 10, 1832. He pulled no punches:
I consider, then, the power to annul a law of the United States, assumed by one State, incompatible with the existence of the Union, contradicted expressly by the letter Of the Constitution, unauthorized by its spirit, inconsistent with every principle on which it was founded, and destructive of the great object for which it was formed.
Jackson threatened the use of military means to enforce obedience to federal laws in South Carolina and, as the other Southern states failed to show solidarity, South Carolina backed down. The more moderate tariff passed in 1833 went a long way to meet South Carolina's grievances, but the issue of states' rights had not been settled.
Later, the question of slavery became paramount, with the issue being whether states had the right to maintain slavery within their borders and whether new territories had the right to choose between being free or slave without interference from the federal government. In the 20th century, "states' rights" were generally a code for segregation.