Barrel saddles have small/deep seats and are light to keep the rider in a position where you can ride at fast speeds and around the barrels without falling off.
Roping saddle have a very thick and solid pommel with a tall and sturdy horn for roping cattle. They are quite heavy and have 2 cinches instead of one to prevent the back of the saddle from being lifted when you tie a cow.
Trail saddles are roomy and has the comfort of both horse and rider in mind over long distances. They have shorter horns and lots of hardware to hold saddle bags and equipment.
Pleasure saddles keep the rider in a classic western position. Most of them are relatively plain looking and have a short horn as well.
A pleasure or trail saddle are best for trail rides.