A crupper is a strap that runs back from the saddle and is then looped under the horse's tail.
Some styles of harness include a crupper. Cruppers are also used on riding horses that have a low or flat wither, to prevent the saddle from slipping forward. This is most often an issue with ponies. Cruppers are also seen on donkeys and mules (although in really rough terrain, saddle breeching is used). Some outfitters use them on horses in rough terrain.
Because the crupper goes through a particularly sensitive part of the horse's anatomy (the skin under the tail is very soft), horses have to be trained to use one and they have to be kept very clean and well oiled or they will annoy the animal. A lot.
Source: Una Smith. This horse is set up for endurance racing.