Graduated neutral density or split filters are handy for scenes with extremes of dark and light. Sunrises and sunsets are examples. In a sunrise, the sunlight on the horizon can overpower any clouds in the sky. The graduated neutral density filter allows less of the intense light to pass through that part of the filter. By using this filter, clouds in the sky are better defined. The second part of the filter brings out the details and colors in the foreground. Without the filter, the foreground of a scene with a sunrise or sunset is usually dark and shadowy. That part of the filter allows more light in to enhance the foreground.
Polarizing filters are for taking photos outdoors in bright light conditions. Sunlight on water, ice, glass, or other reflective surfaces can cause glare. This filter reduces glare and can alter the color of the sky and water to make them a deeper blue. Outdoor objects like a wooden deck chair or leaves or berries on a shrub reflect light. Because of this reflection, these objects will have duller colors in an unfiltered photo. With a polarizing filter, the small amount of reflection from things like berries or leaves is reduced, making the hues more vibrant. Polarizing filters are much like the lens in polarizing sunglasses. The filter can be rotated until the aspects of the scene have the depth of color desired.
A UV (ultraviolet) filter is clear and one that many photographers will leave on their favorite camera lens to protect the actual lens from scratches. It can be more useful than that, though. Without a UV filter, foreground images especially in higher altitude locations can have bluish casts and not be as sharp. A filter prevents some of the UV rays from entering the lens. A UV filter can also reduce some of the haze in a distant landscape.
Dust and other particulates in the air cause objects in a photo to appear hazy or unclear. Haze filters have a yellow tint which reduces these effects. They do not reduce glare like a polarizing filter.