Confederate army was the name. They were also called Rebels too.
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People who were not in the army were called civilian.
The Northern, or Union Army
The Union army (or armies)
The Army of Northern Virginia was the principal Confederate army in the Eastern theatre. It was led by Robert E. Lee after the serious wounding and hospitalising of Joseph E. Johnston.
During the US Civil War, the Northern forces had a significant advantage in terms of the available number of men for the army.