A pelham bit is a bit that functions with the capability to work like a snaffle bit or a curb bit in a horse's mouth. A pelham bit is basically a plain dressage-legal snaffle bit, but with a second bit connection at the end of a shank. This shank, which creates curb action, makes this bit illegal for dressage competition.
A pelham bit is often viewed as a harsh bit, but when used with double reins in educated hands the pelham is actually a very humane bit. Contact should be taken on the snaffle rein, and the curb rein should be left slightly slack. The curb rein should not be engaged at all times, but it should also not be left to flap with each stride- which can be distracting and painful for a horse. Pelham bits are sometimes fitted with converters or "roundings" which allow a rider to use a single rein with a pelham bit. In this writers opinion converters are never acceptable as they turn a pelham into the equivalent of riding with contact in a western curb. If one needs the control of a curb bit at all times, the finessless Kimberwick is a more appropriate choice for an english rider.
The pelham bit has been popular with show jumpers for some time but is increasing in popularity as a bit for schooling dressage. The benefit to schooling dressage in a pelham bit is that the horse can be collected on the snaffle rein, but the shoulders can be lifted and gaits improved with judicious use of the curb rein. The pelham is sometimes used to prepare a horse for working in a full double bridle.
Addendum: In some organizations, a pelham bit is allowed in upper levels, but only on ponies who lack sufficient space in their mouth for a double bridle.