Regarding IR filters....

IR filters are not built into the camera.

I suspect these are what the previous response was referring to.... There is a UV filter built into digital cameras, in most cases. And to do some types of IR photography, the filter over the camera sensor is removed or replaced.

But IR filtration is a whole different thing than UV, giving other-worldly effects in images. Take some time and study IR photography because it often involves a combination of camera modification, filtration and post-processing. There are also some ways to simulate it with post-processing. There are different methods and filters, depending upon the "look" you want in your images.

I saw a recent post about using an IR filter to cut through haze... don't know a lot about that.

Again, I suggest you research IR photography, if that's your interest. It's fairly specialized, there are several different "looks", and different methods are used to achieve them. For more info, you might want to check out:

http://www.lifepixel.com/?gclid=CICzwfSbm8ICFQeVfgoduq8Aig

Those guys do the camera conversions (NFI) and have a good, informational website.

Filters for black & white...

If your intent is simply to make black and white images (not the IR "look" ), you really don't need filters for that at all. We used to use orange, red, yellow, green filters with B&W film... but with digital they are unnecessary, as you can more effectively apply filtration in post-processing.

Color correction and color conversion filters are another type we used a lot for film, but are also pretty much unnecessary today, too, with digital. I'm more likely to put a filter over a flash, to make it better match other light it's mixed with, than to use any of the dozens of different CC filters on my camera.

ND filters...

FYI an ND 0.3 filter cuts one stop and an ND 0.9 cuts three stops.

Uses for ND filters vary, but usually are to allow using slower shutter speeds and/or larger apertures for brighter daytime shots... such as when you want to blur moving water or are looking for a shallow depth of field effect. It's usually less necessary to have fine control of this, partial or single stop filters are often most useful for videography, where there is less latitude in shutter speeds.

If this is what you are wanting to do, I'd recommend starting with a 6-stop or 8-stop ND filter. This will usually put you in a useful range for still photos, that can be fine tuned with the ISO, shutter speed and aperture.

A use for a weaker ND filter is to deepen the blue of the sky and make clouds "pop" when shooting with an ultrawide lens, that would see uneven effect if a polarizing filter were used. Speaking of which...

Circular polarizing filter...

... is arguably the single most useful type of filter for digital photography. While many other types of filters can be closely emulated in post-processing, many of the effects of a C-Pol cannot.

There are linear and circular polarizers. Autofocus cameras in general require the circular type (linear will interfere with the autofocus).

C-Pols are used to darken the sky in scenic shots, reduce reflections off water, glass, etc., increase saturation and contrast, and more.

Graduated ND filters...

Are partially gray and partially clear. This type of filter is used to balance brighter sky with foreground, and similar. Usually one, two or three stops, it's usually best to use the rectangular type that slide into a holder mounted on the front of the lens. This allows you to move the gradation around as needed, to match with where you need the effect in the image. (Screw-in ND Grads force you to position the horizon in the center of every image... not a good thing.)

Frankly, I have a nice set of ND Grads that are pretty much just gathering dust. I can do as good or better job in post-processing, combining two shots made at different exposures, or with double-processing of a single RAW file. This works better than a filter because I can use layers and masks to make the effect exactly match my image.

Also, many post-processing softwares have graduated filter effects available, which can be pretty effective.

There are many good filter brands. Personally I use some B+W, Hoya, Lee, Kenko and a few Cokin. I probably have some others I'm forgetting. Marumi is another brand I've heard good things about recently.

Nearly all manufacturers make different grades of filters, offered at various price ranges. For example, last time I looked Hoya had five or six different grades of C-Pol and B+W offered four or five.

Look for high quality glass and especially for multi-coating. This is important as it will minimize loss of image quality, and keep flare from occurring, as best possible.

Some filters use brass mounting frames, to minimize binding in the filter threads on some lenses. However, many lenses today use plastic filter threads (most Canon do), that are less prone to binding already.

Also look for thin mounting frames. There are special, thin ones to prevent vignetting on ultrawide lenses, but they cost a lot more and some don't have front threads (so you can't stack another filter on top and they need a special lens cap, if you leave them on your lens).

I've found slim filters unnecessary even on my widest lenses, by using C-Pols (which are two layer filters and thicker than most) from B+W, whose standard frames are pretty darned thin. These also are brass mounted... and I use their MRC multi-coated and Kaësemann MRC C-Pols, which are among their very best (they now have one grade higher "MRC Nano", with enhanced multi-coatings... more scratch resistant and easier to clean).

Most quality filters in even the largest sizes really aren't all that expensive... and we need so few today with digital... that I don't think it makes sense to buy cheap and poor quality. Remember, every shot you make through that filter will be effected by it... that can be a good thing, or a bad thing.

I am not a fan of step-up ring adapters either, allowing using oversize filters on a lens. I used to use these with film cameras, because I had literally dozens of filters and it was cost prohibitive to buy all of them in all the different sizes. However, with today's digital you really only need a few filters, so I buy them in all the sizes I need. (I also consider filter sizes when I buy a lens, preferring to keep to a few common sizes whenever I can.)

One key reason I don't like step-up or step-down rings is because they make it difficult or impossible to use a proper lens hood. I always use a hood... and it's even more important to use one when also using a filter. By using the "correct" size filter, the proper lens hood is also usable.

You might be able to be a bit selective about sizes... For example I use a C-Pol a lot more often on wide angle lenses, than on telephotos. So if I were buying today, I'd get the size to fit my wider lenses first.

An IR filter is not required for haze reduction because, for one thing, haze is the result of higher frequency light waves (thus more towards UV).

Digital cameras have an IR filter (of sorts) built into them because the imaging chip in its native state is particularly sensitive to IR wavelengths of light - thus the manufacturer has an IR cutoff filter layered over the CCD or CMOS chip to enable human-visible light to get through more cleanly. When you send your DSLR in to "convert" it to an IR camera, they service removes this filter thus enabling the chip to "see" those frequencies.

An IR filter that screws onto the front of your lens (like an UV or CP filter does) is one that only allows IR wavelengths through (thus it appearing pretty much black to our eyes - since we do not see those wavelengths). Using one of these on a non-converted DSLR will approximate the effect of making the (permanent) conversion described above, though of course it will require longer exposures since the IR cutoff filter on the chip will be removing much of the IR light passing through the filter on the lens.

In any case, IR photography (film as well) is "black and white" because IR light is outside the band of wavelengths we can see - from the shorter wavelengths that are deep purple to the longest bands that are red - IR are even longer wavelengths that fall outside of the entire RGB rainbow concept altogether. There was a technique in film days of generating "false color" in IR shots, pretty much by assigning different visible colors to different wavelengths in the IR band, but of course those colors have nothing to do with what we normally perceive.

IR shots of nature scenes show the leaves of trees (that we see s green) as being bright or all white since those leaves reflect most of the IR wavelength light that falls on them.

Consider that any surface placed on the front of your lens will bear on the performance of the lens. Using a filter could cost you quality in your images.
Today photographic filters are better than ever but try to buy the best that you can afford. Hoya makes excellent filters and the big brands like Lee, Sing Ray and others make excellent but expensive filters.
My advise to you is do not use a filter unless it is absolutely necessary but if you have to use one make sure you are using the best that you can afford.