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There are two main types of neutral density (ND) filters: graduated and non-graduated. Graduated ND filters are darker at one edge and lighter (usually completely clear) at the other. They're typically used to balance out a bright sky with a darker foreground, e.g. in landscape photography. Their usage is dealt with in detail in this question.

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For long exposure work you'll typically want to use a non-graduated neutral density filter, to ensure the same effect across the whole picture. In short, the ND filter works by reducing the amount of light coming into the lens, so you can compensate with a slower shutter speed than would otherwise be possible.

Personally I would buy a CPL(circular polarizer filter) and use that when possible since it is dual purpose. It will block some incoming light and allow a wider aperture, but also give some pop that CPLs are known for.

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You will want a neutral density or ND filter. It essentially darkens all parts of the image giving you a uniform exposure that is darker then it would be otherwise. Depending on the camera, some have a built in ND filter, or some accept a filter accessory that you can screw on to the lens or drop into the filter slot.

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Let's say the camera is taking a well exposed picture at ISO 100, f/8 and 1/100s. Make a mental note of those settings. Now you can start preparing for the long exposure shot.

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Note that a uniform ND filter does not change the Dynamic Range of the scene, as the bright areas get darker in the same amount as the dark areas.

I'd like to be able to take long exposure photos during daylight (like the examples in this Flickr group). I'm told that you can do this by using a neutral density filter. So my question is: what is a neutral density filter exactly, and how is it used to achieve that smooth long exposure effect?

Another type of ND filters is the graduate ND filters. These filters usually attenuate the light only at one half of the frame and gradually get clearer on the second half. These filters are used when shooting landscapes, where the sky is much more bright than the scenery and using good exposure for the ground will overexpose the sky. A graduate ND filter lets one reduce the brightness of the sky and get it captured nicely with the ground.

A Neutral Density (ND) filter is a filter that reduces the amount of light captured by the camera evenly across the visible spectrum. As such, it looks grey to black (depending on the filtration power) and does not cast color on the received image (like blue or yellow, e.g., filters will do).

We have lots of questions that cover some more specialized questions on neutral density filters that may interest you. See:

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There are some (expensive) variable ND filters. These are made from two polarizers, where the front element rotates and thus let you set the amount of light being cut (up to virtually no light passing through).

Luckily there's a handy conversion table on Wikipedia, and better still you can easily get by ignoring everything but the shutter speed multiplier.

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When using a ND filter, there is a need to compensate for the reduction in light by using longer exposures (or higher ISO, but it usually defies the intention of ND filtration). This way, one can capture long-exposure images that are not possible w/o the filter as the camera (set to its lowest ISO and possibly smallest acceptable aperture) is at its slowest speed for a good exposure.

ND filters are categorised by how dark they are (i.e. by how much light they block out). There are a few different numbering schemes for describing this darkness which can be confusing to the first-time buyer:

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Examples for such situation is when shooting waterfalls in a fairly lit location. Then, the speed for correct exposure is not slow enough to create the required streaming water effect. Using a ND filter, one can reach slower speeds, as if the scene lighting was dimmer.

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ND filters are marked in multiplies or stops. An 8X ND filter is similar to ND3. Both cut 3 stops of the light. If the camera's meter tells that 1/500 sec is required for the given aperture, then using a ND3 filter will allow an exposure of 1/64 sec with similar outcome.

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"Neutral density" just means that the filter is a pure shade of grey: it shouldn't (if well manufactured) add any colour tint to your photographs.