We're starting to see gag bits used on Icelandic Horses. The mechanics of this bit are not very nice to the horse (or the horse's mouth). This bit is usually used in place of training or good / natural horsemanship, and can be used to force gait. A good, knowledgeable trainer won't use one.
They are also known as: gag bits, wonder bits, Dutch or Continental bits, and Pessoas.
The gag bit works on the horse's lips and poll simultaneously. The pressure on the lips tends to make the horse raise its head, which is useful for a horse that tends to overextend.
Gag bits are used mainly for horses that are strong pullers or for horses that need retraining... mainly for increased control at times when a horse may try to run off with the rider.
They are not permitted at any level of dressage, since dressage riders are trying to get the horse come down onto the bit, and want to encourage the horse to accept contact. Additionally, the horse is supposed to be completely submissive in dressage, and a gag bit gives the impression that it is not. Gags are also never seen in the hunter arena, again because they wish to portray that the horse is an easy ride, and because they want a long, relaxed frame with the neck stretched out, rather than a high neck.
You will also see that the Icelandic Horse was shown to Al Roker in a television interview with it's mouth strapped shut and a gag bit: