Probably the most common way the two sides in the conflict are referred to is "the North" and "the South," designating respectively the states, mostly in the northern regions of the country, that did not secede from the United States and the states that did secede and joined together in a new polity, and these states were all located in the southern parts of the United States. The southern states called this new union they had created the "Confederacy," or more formally "the Confederate States of America" (a.k.a. the "CSA"), whereas the northern states that remained united under the U.S. Constitution were called collectively "the Union."
Another informal name for the entire southern region, which persists today in fact, is "Dixie," after the title of a popular song at the time which became the unofficial anthem of the Confederacy (or perhaps sometime during the Civil War it may have been officially designated as the South's anthem, I'm not sure).
In the actual armed conflicts of the Civil War, the two sides had numerous nicknames for themselves and each other as a group and individuals, e.g., for Union troops "Federals" and for the Confederates "rebels," "rebs" or "Johnny reb" for an individual Confederate soldier. Robert E. Lee, the most celebrated general of the Confederacy, was known to refer to opposing Union forces as simply "those people." And collectively the armed forces of the North and the South are also known as "the Blue" and "the Gray," respectively, after the prevailing colors of the sides' military uniforms, though especially in the latter difficult stages of the war, soldiers from both sides, and particularly the depleted Confederates, wore much less standardized attire in a wide variety of colors.