It is a piece of cotton or linen fabric attached to a soldier's headgear that protects the back of the neck from sunburn. It is named for the sun-sensitive British officer who invented them.
Havelocks were not Union issue in 1861, or ever. But, someone decided they were necessary at the beginning of the War, and loyal Union women got together to make them in great numbers for Union soldiers.
Apparently, long-haired Southern boys didn't need them, or no one thought to make them, but they were a Union phenomenon, certainly.
Only problem was--the soldiers hated them! Except for the Fire Zouaves, who wore them proudly. As firefighters, they probably understood the comfort a wet cloth can give to the back of one's neck, plus they had, initially, shaved their heads.
So, here is a drawing of loyal Union women cranking out havelocks in 1861. I remember, as a loyal Union woman, doing just exactly the same thing for the beginning of the 125th CW celebrations, and I am going to assume that another generation of loyal Union women did the same thing last spring, for their brave reenactors.
With probably the same results--no one wore them, except the Fire Zouaves.