To determine how to govern itself. A state is (supposedly) a more unified group of people, and thus those people's local representatives (supposedly) better represent their interests about how to handle everything other than interstate trade and mail service. In the U.S. A modern example of this debate is in gun control and minimum wage laws. On top of the actual substantive policy issue, there's the issue of whether those (and almost everything else in the U.S.) should be decided by the federal government or the state government.

This is a bullshit argument because the Constitution very clearly made Congress the ultimate authority on the regulation of the slave trade. There was no "state's right" for a state to ignore federal laws on slavery. For 50+ years, Congress had been passing laws to regulate slavery and the South was right on board with it, trying to impose their will on the northern states, and the federal government was enforcing pro-slavery federal laws even in states that had outlawed slavery at the state level. (The Fugitive Slave Act being the prime example.)

Now that the power dynamic had shifted, and the Congress was going to be filled with anti-slavery Republicans who might pass a law phasing out slavery over time, the Confederates decided that they didn't want to honor the law and the Constitution anymore. So they committed treason by opening fire on the U.S. Army in April 1861.