What did the Royalist fight for in the Civil War?

("US Civil War" is a contradiction in terms, and there were no "royalists" in it as such). I presume you mean the English Civil war. There the Royalists fought for the "old order" (the king). Cromwell & co objected to near-autocratic rule, finished up winning and bringing in a short-lived republic which turned out worse, so Royalty was invited back.