Many digital cameras have something called AEB, or Automatic Exposure Bracketing. You click the shutter once, and the camera takes 3 pictures instead of just one. In my case I usually shoot a normally exposed image, then one 2 stops under, and another one 2 stops over. If your camera doesn't have AEB you can do this manually.
Photomatix Pro combines the three exposures you've taken into one, and tone maps the image into a JPEG or TIFF that displays a really wide dynamic range. The result is often a stunning picture, with little loss in the shadowy areas, and no blown-out highlights. It's hard to do if you are shooting people because they will move between the 3 exposures, but works great with landscapes. A tripod is recommended, but I've done just fine handheld and Photomatix Pro will automatically line up (register) the 3 images.
Words can't describe the difference, so I've included a few examples to look at. There are other Mac apps that do similar things, but I've had the best luck with Photomatix Pro. They also make a Photoshop plug-in and one for Aperture, but I think the full program is the best. Newer versions of Photoshop also support the creation of these HDR images, but the results are not as spectacular in my view. There is also a free, basic version of the program that lets you combine 2 images, but it is simply not as effective.
You can search for a lot of information on HDR photography on the web. I have found this site to be very good, and it reviews the different software that is available. It's very easy to overdo the look of HDR images, and Photomatix Pro gives you lots of control. You can make your images really good, or outlandish. The control is in the hands of the photographer. To many, HDR will be old hat, but a lot of photographers haven't tried it, and will be pleasantly shocked at the results.
Since you can download Photomatix Pro for free, if you are serious about digital imaging, I'd give it a tryout. If you buy the full version it is US$99.00.
Here are some examples of how HDR imaging works. Of course these images are reduced in quality and resolution, so they only hint at the differences.