What is a border states?

In the American Civil War, the border-states were the four slave-states of the Upper South that had voted (narrowly) to stay loyal to the USA. They were Missouri, Kentucky, Maryland and Delaware.

At the beginning of the war, Lincoln's biggest worry was that these states would fall into the arms of the Confederacy. This was why he allowed them to continue practising slavery, so as not to upset leading slave-owners in these states.