The trailbike category consists mostly of bikes with about 4 to 6 inches (100 to 160 millimeters) of travel. While they are designed to climb hills very efficiently, they are generally heavier and a bit more stout than the typical cross county mountain bike. They can handle a lot rougher terrain as well.
Trail bikes are what most people should be riding. They are an excellent balance between efficiency, comfort, and control. Trailbikes are light and efficient enough to get you to the top of the hill, have soft enough squish in the suspension to keep you isolated from rough terrain, and have enough travel to suck up the bigger hits that can leave a cross country racer tasting the dirt.