Growing up in Tennessee, our middle-school US history classes told us that the Civil War was started over states' rights. We were too young to ask "states' rights to do what?", and our teachers never volunteered an explanation. Fortunately, the Confederacy itself made clear exactly which rights they were interested in.
From the South Carolina declaration of secession: "an increasing hostility on the part of the non-slaveholding States to the institution of slavery, has led to a disregard of their obligations, and the laws of the General Government have ceased to effect the objects of the Constitution"
From Mississippi's causes of secession: "It refuses the admission of new slave States into the Union, and seeks to extinguish it by confining it within its present limits, denying the power of expansion.
It tramples the original equality of the South under foot.
It has nullified the Fugitive Slave Law in almost every free State in the Union, and has utterly broken the compact which our fathers pledged their faith to maintain.
It advocates negro equality, socially and politically, and promotes insurrection and incendiarism in our midst."
From the Constitution of the Confederate States of America: "The citizens of each State shall be entitled to all the privileges and immunities of citizens in the several States; and shall have the right of transit and sojourn in any State of this Confederacy, with their slaves and other property; and the right of property in said slaves shall not be thereby impaired."