I hunt both in the daytime and at night. With hunting coyotes at night, I usually hunt them basically the same way I do fox.
It can be a real adrenalin rush when you see those eyes coming in and you can't do anything. It really kind of piques the excitement.
Most of us will skin the animal out and then sell the furs on the fur market just like a trapper would do. Some guys, if they get an especially nice animal, may get a mount done. Some guys are into tanning their own furs.
One method I messed with this fall is using a headlight, like a nightlight that coon hunters wear, that goes on your head or hat. What you do with this technique is scan back and forth with the headlight using a red lens. You can pick up the eyes quite a ways out there.