Southern states were concerned with protecting Slavery. The chief political question of the Presidential election of 1860 was whether slavery should be extended to the new territories of the United States. Democrat Stephen Douglas believed that the territories should decide themselves whether slavery should be allowed or not. Republican Abraham Lincoln indicated that slavery should not be allowed in the territories. Southern Democrats, who believed that Douglas' position amounted to capitulation to Northern abolitionists, formed a splinter party called National Democrats and nominated John Breckenridge, whose position was that slavery must be extended to the territories.
Thus, the Southern political activists who split from Douglas' ticket were against state's rights, not for them. They opposed granting territories the right to decide how they were governed. State's Rights were a favorite canard of Southern politicos at the time, and even among Confederate apologists today, but (to be generous) the facts do not support their position.
The precipitating event of the Civil War was the election of Lincoln. Although Lincoln's platform, and all his speeches (and indeed, his policies until the outbreak of the war) stated that he did not believe the President and Congress ordinarily had the power to regulate slavery where it already existed, in the States, the pro-slavery politicians heralded him as a leader only of the North and stated that his policies amounted to radical abolitionism. Buoyed on such rhetoric, several states appointed "secession commissioners", who consulted with other states on the possibility of leaving the Union, then held "secession conventions" at which they declared that they would do so. Lincoln held these conventions to have no legal effect. When he decided to resupply Fort Sumter, a federal military facility on an island near Charleston, South Carolina, the South responded with force and the war was on between the Union and the Confederacy.
The Union won. Although there were several close calls that might have prolonged the war greatly, two factors would have led to the Confederacy's inevitable defeat: first, their total dependence on slave labor which the war largely denied to them even before emancipation became a specific war aim, and second, the industrial might of the Northern states. Swept up in the general passion for secession, however, these factors were ignored by Southern leadership.
As Bruce Catton wrote, the one group of people in America who knew exactly what the Civil War was about from the very first shot were slaves. They knew that the Civil War meant their freedom long before any soldier ever knew.
The Civil War was enormously destructive. More Americans died in one day at the battle of Shiloh than had died in all other previous American military actions combined, including the American Revolution and the French and Indian Wars. Corruption and war profiteering was rampant both in the Union and in the Confederacy. And the actual battlefields themelves, as well as many Southern cities, were physically devastated.
The Civil War was followed by a period known as Reconstruction. It also resulted in a dramatic growth in the federal government. Political tensions between the South and the North dating from before the Civil War remain relevant to the political environment, even today.