Partisans can appear in states where they were historically active if the enemy controls most of the territory but with too few troops. Spontaneous appearances of partisans, made up of small bands of men, will then occur in the wildest regions of the area. In game terms the units are small and not very fit to engage in open fights but have an excellent hiding capacity, can launch ambushes and move very fast, all without the need of a leader.
As partisans, they get heavily penalized if they leave their respective home states. You can use them to retake an enemy town, or most interestingly, to burn a depot or stockpile. You can keep them in hilly terrain or wilderness regions so that they remain hidden and then have them jump on unexpected supply wagons (transported by rail or moving by normal mean), thus crippling logistics assets at an opportune moment!
As the opponent of partisan attacks, you will have to patrol the area, engage the enemy (but regulars are penalized in wild terrain!) or at least guard important locations. Since partisans are very mobile, they can often escape a fight even if in the same region as an enemy, so be sure to bring fast units such as cavalry or rangers (if you have some to spare). They will do marvels in counter-insurgency operations!

Confederate bushwacker.