The white men of the Confederacy elected their own president, Jefferson Davis (but the enslaved African-Americans were still not allowed to vote, and neither were women or anyone who didn’t own any land). Lincoln decided to use the United States army to recapture the South and force them to be part of the United States.
The actual fighting started in 1861. The Confederate army was greatly outnumbered, and the United States also had more guns and more ammunition for their guns. The only hope of the Confederates was to get help from England or France, the way they had in the American Revolutionary War a hundred years earlier. But this time both England and France said no. They would not help any country where it was legal to hold people as slaves. England, which really wanted a lot of Southern cotton for their cotton mills, did send some guns and ammunition to the Confederacy, but no soldiers.
But the Confederates and the United States fought a lot of battles anyway. Hundreds of thousands of men died in these battles, without anything being solved. Many other men lost their legs or their arms or their eyes. This was mainly because of the recent invention of cheap steel. Using that steel, people had invented new kinds of guns which were more powerful and more accurate than earlier guns. Using these new guns, it was a lot easier to kill somebody. But generals kept on fighting battles in the old way, with a bunch of men marching along in a long line. The two lines would just shoot at each other until most of them were dead.
In the end, the Confederacy ran out of both men and guns. Robert E Lee surrendered to the United States general, Ulysses S Grant, in April, 1865. So the southern states had to remain part of the United States, but not willingly.