A conflict between the southern states of the United States of America which declared that they had seceded in 1860-1, and the remaining loyalist states, taking place 1861-5. Depending on who you ask, the issue was either states' rights vs. the power of the federal government and/or the fact that the southerners refused to end slavery.
The idea that the southern states could win the war and become independent as the Confederate States of America is such a cliché nowadays in AH that Dr Pervez Hoodbhoy memorably described it as “the most original What If? ever”.
The ACW, along with World War II, is probably the area of history that has been most 'Podded', i.e. the result of almost every battle and skirmish being changed has been considered at some point or other.
In interesting discussion of the history of CSA-victory stories is found here.
The Worst American Civil War Alternate History Cliché
Common settings of alternate history works
American Civil War Points of Divergence
United States of America Points of Divergence