Photographing fireworks is something that sounds intimidating, particularly to the newbie photographer. In reality, it's not that hard to get great results, with just a few pointers, once you know the technique.
The proliferation of digital SLR's has made shooting fireworks easier than ever. You can see your results immediately, allowing you to correct any exposure or composition errors along the way. Digital also allows you to take lots and lots of shots, without worrying about the price of film and processing, or missing a particularly spectacular burst because you were changing rolls.
This tutorial applies to digital SLRs. Unfortunately, I have never tried to photograph fireworks with a P&S camera, so I can't give much in the way of advice.
Shooting fireworks doesn't require any special equipment. Most amateur photographers should have everything on this list. Anything that you don't have already is worth picking up, since it will be useful in more situations than just fireworks.
digital SLR (I use a Canon EOS 5D)
wide-angle lens (I usually use a Canon EF 17-40mm f/4 L)
good, sturdy tripod (absolute requirement! Exposures will range from 2-10 seconds)
shutter cord or remote
memory cards, memory cards, memory cards
charged, spare battery
The lens you need really depends on the size of the fireworks display, and how close you are. Often you'll be surprised how wide you need to go, so you should go a little wider than you think you'll need. I always try to be close to water if I can, to catch reflections, which usually requires my widest angle lens, even on my full-frame 5D.

Arrive at the show very early, so you can scout out a good spot with no obstructions. You are going to be shooting very wide, and it's hard to predict where the wind is going to carry the bursts. Be very observant of tree branches, fences, poles, overhead wires, etc. Once you've found your spot, plant your stuff, and stay there so you don't lose it (it helps to have a friend or two along, so you can switch off guard duty).
Setup your camera on the tripod and decide on a rough composition. Try to include some ground elements, to provide scale. You won't be able to finalize your composition until a few minutes into the show, when you can determine exactly where the bursts are, and how wide you need your lens (you may be surprised just how wide you'll find you need to go). Pre-focus your lens to infinity, and then set the focusing mode to manual focus (very important!)
You should set your camera to manual mode. To start, set your ISO to 100, aperture to f/11, and set your shutter to bulb (if you're using a Nikon, set your aperture to f/16 to compensate for the ISO 200).
If your camera has a noise reduction mode, where it takes a black frame after a long exposure, you should turn this feature off. If you leave it on, then you will miss out on some shots, while you wait for the camera to take this black frame.
I recommend that you shoot in RAW mode, to make adjustments easier later.
Now, just wait for the show to begin.
Aug'06 Update: at this year's Sound of Light Fireworks Competition, I've been shooting at f/16 instead of f/11. I've found that the colours of the bursts come out better this way. But, dimmer colours, such as gold, don't show up as well, so it's a bit of a trade off.

It's time to start shooting. I use my shutter cord to open the shutter as soon as I hear the rocket launch. I hold the shutter open anywhere from 2 to 10 seconds, depending on how many rockets are in the air, and how bright they are. I also try to build a mental image in my head, and release the shutter when I have both high bursts and low bursts, so the frame is filled. When I think I have a mental image of a good composition, then I release the shutter.
As you're shooting, periodically look at your LCD and your histogram, particularly near the beginning. Make sure you're capturing the full bursts (not cutting off at the edges), and that you're not blowing out too many highlights (expect the center of the bursts to be a bit blown out). Adjust your composition and/or your exposure accordingly. If you're blowing out too many highlights, reduce your aperture by 1/3 or 1/2 a stop. If too dark, open up by 1/3 or 1/2 a stop. Do not try to compensate the exposure with the shutter times, it won't work.
Take lots of images, it is digital after all.
Process your RAW images as you normally would, except you should expect to do a little more cropping than usual. To reduce the appearance of smoke in the image, use the contrast slider, and pull it to the right. You'll be surprised just how far you can increase the contrast, without having a stong affect on the image, other then the disappearance of the smoke. In some cases, you'll find you can pull it all the way over to the extreme right.
UPDATE: Another way that I've found to reduce the appearance of smoke, and to really make the fireworks pop, is to use the black point slider in the levels adjustment. By moving the slider towards center, you will darken the sky, and reduce the smoke. I had good results with this method for this years
Sound of Light photos.
That's all there is to it. It's that simple. With just a little bit of practice, I'm sure you'll be shooting fireworks that you'll be very proud to display. And if you do, please post a comment below, with a link to your gallery!