There are many different types of saddle pads for english saddles. The common types of english saddle pads can be divided into three categories: corrective (usually used in addition to and/or covered by another saddle pad), shaped pads (The type of english saddle pads cut to saddle shape). Shaped pads are frequently referred to as contoured pads. Contoured english saddle pads, preferably in fleece or sheepskin, are "correct" for hunter classes and english pleasure. The third type of english saddle pad, "Square" pads, are popular for dressage and permissible for jumpers, cross country, and local circuit english pleasure classes.
Either type of english saddle pad can be used alone, or paired together, one on top of the other for various reasons- though for most horses that's too much padding.
Of course, riding at home you can use any kind of saddle pad as long as your horse is comfortable. For practicing and pleasure or trail riding they make shaped dressage pads and square A/P pads so it's just a matter of taste for most riders. Square english saddle pads, or even swallow tailed square pads are currently in fashion for training rides in almost all english disciplines.
There is also a variation on the square saddle pad called a "baby pad". Baby Pads are extremely thin and offer little to no cushioning- Because they are easy to wash and low maintenance, they are used underneath more expensive shaped pads to keep them clean.