A neutral density filter (also called a fader) will darken all or a portion of the image without any color shift. Neutral density filters have 3 types: fixed darkening amount, graduated (darker on top, less dark on the bottom, for example), or may be variable (a twist of the filter makes the entire image darker or lighter).
The fixed amount ND filters will darken the entire picture one or more stops (or portions of), causing the camera to require a longer exposure to capture the image. Multiple fixed ND filters may be combined to make it darker than a single filter. Making moving water 'smooth' or 'silky' is a prime example. Pre-1960 movies used fixed ND filters to shoot nighttime (appearing) scenes in daylight, as the films were too slow (low ASA/ISO) to be used at night. Graduated ND filters with a darker top are frequently used to photograph sunsets with subjects 'below' the sun that need more exposure. The best types of these filters are square or rectangular and may be moved up and down to move where the horizontal light/dark transistion occurs in the picture before the shutter is opened. Variable ND filters do the function of multiple fixed amount ND filters, with any amount of darkening desired, all in a single filter.
Polarizing filters, on the other hand, although they appear to be made from darker glass, they are not. Instead, it is blocking all the light perpendicular to the polarization. Its biggest use is to eliminate reflections from smooth surfaces like glass or water. They also have a tendency to darken the sky a bit making it more 'vibrant'. Circular polarizing filters are rotated on-camera (lens) to achieve the 'right' amount of polarization. Achieving the desired effect is dependent upon where the camera is, the reflection is, the angle of the camera sensor in relation to the reflecting surface, and, the rotated 'directional position' of the polarizing filter.
ND filters and polarizing filters are the most commonly needed filters in digital photography. UV and clear 'protection' filters are a complete waste of money and may cause unwanted image degradation, ghosting, filter-flare, bluriness, and even lens damage if broken.
Like most photographic equipment, quality does not come cheap. In general, filters costing less than about $100 (US) frequently lack the better coatings needed to prevent ghosting, flare, etc. The cheaper filters of any kind will usually cause the most image degradation because of the lack of quality materials/coatings/manufacturing. Some manufacturers such as Hoya produce multiple lines of the same filter types at various price levels, with better quality filters progressively more expensive. For example, at my favorite online photography equipment store, Hoya 72mm fixed amount ND filters vary in price from $32 to $100. Various manufacturers 72mm variable ND filters vary in price from $35 to $383. Square/rectangular ND filters from one manufacturer vary in price from $20 to $300...plus a one-time cost of filter holder and adapter, usable with all their filters. ALthough filters are less costly than lenses, one does not want to be forever buying better and better filters because of problems caused by lesser quality filters.
