There are two types of pelhams - the more common english pelham, and the lesser known western or "cowboy" pelham.
It has a similar action to a snaffle/curb double bridle, as seen from Third Level and on in dressage. The snaffle rein on both the double bridle and pelham is for the aid in lateral flexion, where the curb rein provides the aid in vertical flexion of the poll, and also asks the horse to sit more on his hind end. The drawback is that the curb action on the pelham, even when not intentionally used, will come into play slightly because the snaffle and curb are not truely seperate. There is also a disadvantage in that when the mouth piece is jointed, the curb action is greatly altered, and vice versa when it is solid. Shank length can vary from five centimetres to ten centimetres.
The english pelham was originally designed with the intent of being a dressage bit, mostly for riders learning to use double reins and horses moving up to double bridles. However, it was later banned in competitive dressage (by the FEI and national dressage federations), I'm not sure of the official cited reason for this.
The western pelham was primarily created as a training bit, modeled after the english pelham.
The english pelham is legal in jumper rings, eventing rings (cross-country and jumper phases, not sure about the dressage phase), and hunter rings. It is also legal in polo. I'm not sure of the legalities in saddleseat. The western pelham, as far as I know, is not legal in any competitive ring.
As mentioned, both the english and western pelhams are often used as "transition" bits for switching horses from snaffles to double bridles or curb bits, and for riders learning to use double reins and respect of the curb.
People also use it for hunter/jumper horses. Just this past Thursday, my sister and I schooled a hunter mare in a pelham for the first time. We call her "Tank Girl", because she is known for dropping her shoulder and head and tanking around corners and into jumps. However, the curb rein on the pelham got the point across really quickly that she wasn't allowed to do this, and by the end of the lesson she was more adjustable and her form over fences was much better because of it.
The pelham is sometimes used in strange or cruel ways - just as any bit. Decide for yourself what you consider each of these.
In the hunter ring, it has become a fashion statement, much like the standing martingale (though this is not always the case, of course).
Rein convertors are used - most notably in the jumper or eventing ring - to turn the double reins into a single rein. This combines the action or the curb and snaffle together, creating the action of a kimberwicke. This practice is also called "pelham rounding".
Some riders will even remove the snaffle rein completely and ride solely off the curb - they should have bought a different bit!
In polo, it is sometimes converted into a gag bit. I do not know much about this practice.
The western pelham.
The english pelham. This one features a jointed rubber mouthpiece.
A polo horse in a pelham, draw-reins through the snaffle ring.
Eric Lamaze's Hickstead in a rounded pelham.
Hickstead ridden only on the curb. Bad boy, Eric!
Me and Freddy. He did great in a pelham.