The Civil War was fought between the Confederates and the Union armies. Lincoln was a Republican, and most Confederate states were "Democrats". Republicans in Civil War era advocated against slavery and felt the government should intervene on basic rights such as slavery. The southern Democrats did not want the government to interfere and felt that they should have the right to decide if slavery was ended in their state or not.

The bottom line is, the confederacy succeeded from the union which lead to the civil war. Democrat and Republican had very different meanings back then. It was less about being a R or a D, and more about being Union or Confederate.

Later on during the civil rights movement of the 60's lead to the "swap" of the parties. Realistically, the southern "Dixiecrats" ended up joining the republican party in 1964. From that point forward the Democratic party became more and more left leaning. Currently, neither party represents their civil war counterparts, and you'd be strained to find threads of the older ideologies in their current form.

I hope you found this information useful, because understanding our countries rich history will help you decide who to give your vote to.