A Polarizing filter (usually comes in a circular shape) is mainly use to reduce the glare or the reflections from non-metallic reflective surfaces like water or glass. It also reduces the exposure but mainly use to saturate the colors and enhance the clarity of the image. It works best when the camera is on the right angle to the sun light. Polarizing filters are not normally used in wide-angle shots that include sky, as you'll get an uneven sky tone due to the wide range of angles of the light entering the lens.
A Neutral Density (ND) filter is use to reduce the overall exposure uniformly giving an uniform reduction in light across the frame without affecting the color. ND filter is commonly used in a situation where there is a bright sun light and using a wide aperture to get a shallow depth of field would result into a overexposed image. It is probably more popular to use a ND filter when you want to use the slower shutter speed to get the cloud effects or milky/silky effects of the water with a long exposure, which could possibly overexpose the image without the use of a ND filter.
There is another type of filter which you could possibly be confused with ND filters and that is called Graduated Neutral Density filter, also known as a Graduated ND filter, Split Neutral Density filter, ND grads, GND, Split ND or just Graduated filter. It is use to reduce the light over certain part of the frame but not the entire frame, and mostly used by placing it in a holder which gives you the flexibility to darken the area of your choice and leave the other parts of the frame as it is. The Graduated Neutral Density filters are much more common in landscape photography in which you can darken the sky and extract the details while maintaining the correct exposure on the foreground at the same time.
Usually Graduated Neutral Density filter comes in two types and they are:
1. Hard edge
2. Soft edge
A hard edge filter is used when there is an abrupt change in brightness; for example a field with a horizon to a bright sky (good for seascape shots). A soft edge filter has a smoother change from the lighter area to the darker area. This type of a filter is used when the bright and the dark portions are not distinctly separated; for example a frame with a mountain and the sky. A soft edge filter is less noticeable than the hard edge filter, and also, it has a benefit of making the sky more intense by darkening the sky that is closer on the top of the frame.
Even though a polarizer and ND Grads both can be used to darken and add details to the image but they work quite differently. A grad filter works with same effect at any angle to the sun whereas a polarizer filter’s effect is stronger at 90 degree angle to the sun. As I mentioned earlier on the post, a polarizer filter can produce some strange variations in the sky tone when used with the wide angle lenses or an ultra wide angle lens, whereas ND Grads have the same effect across the frame regardless of the lens type. Also, graduated filter only darkens the certain area of the frame which you can vary by changing the upward or the downward direction of the filter that slides into the holder (may not be the case if you use a circular grad filter though).
These days, many photographers try to replicate the effect of a graduated filter by using bracketed exposure shots and post process them using a software, but you can't do everything in the post processing that polarizer filter does on the field. Some of the photographers prefer to add-in a grad effect using a software but if you have already blown out the details in the original photograph, there is no way to get those details back during the post processing. If you do not want to use grad filters at all, you can shoot in a RAW format and take 3 or 5 or more automatic bracketed shots and merge them (and create an HDR image) by using the Adobe Lightroom.