HDR (High Dynamic Range) images have become very popular in the last couple of years thanks to the availability of digital cameras and inexpensive software such as Photomatix and Adobe Photoshop CS2 and above.
What is HDR photography? Simply put it is increasing the exposure range of an image by combining a number of versions shot at different exposures. By doing this you can create one image that displays excellent detail from the darkest shadows to the brightest highlights. HDR does this through a process called 'tone mapping'. For more information on what HDR is and how it works, check out 'HDR : High Dynamic Range explained so anyone can understand' and 'Wikipedia: High dynamic range imaging'.
So what does it look like? Like most new photographic techniques, when you first see an HDR image you will be intrigued and quite possibly, amazed. They definitely have their own special 'look' and when done well can be very impressive. To me they often look more like illustrations than photographs.
Unfortunately, like all techniques, they can be over done and over used and because the effect is so powerful, when they are done badly they look really bad. I am sure if you have a look around the internet you will be able to find lots of great examples of how NOT to use HDR, however, I would like to show you some examples of how it should be done.
Underground Station:

Boat:

Old Movie Theatre:

Misty Mountain Road:

Red Chevrolet:

Snowscape:

Sunset:

Train:

Cityscape:

Golden Gate Bridge:

As you can see the effect can be quite amazing. There are lots of sites where you can see HDR photography, both good and bad. Here are some to get you started:
HDR Creme
Flickr - Best of HDR Pool
HDR Tutorial Round-Up
Good luck and remember, use it wisely...