HDR or High Dynamic Range Photography is essentially taking a combination of images and merging them into one photo in order to create a range of light that is otherwise impossible to attain from a traditional camera sensor. The benefits of this are vast, but most simply, when done well, it allows you to create a more accurate depiction of what a given scene looked like at the time the photograph was taken.
If you are taking a photo of a sunset it's nearly impossible to expose for all of the elements of the scene. You can't have the highlights, shadows and mid tones all come out correctly simply due to technological limitations. With an HDR photo you can. By taking correctly exposed shot and merging it one with one that is overexposed and one that is underexposed you are able to take the full dynamic range and turn it into one beautiful image.
Another benefit of HDR photography is that it allows you to be more artistic. Many will argue this is an HDR photo's fatal flaw because there are so many people who take bad HDR photography. Either it's too surreal, too saturated or a combination of both; there's much more bad HDR out there than good. However, if done well you can create beautiful scenes that are much more vivid three dimensional than an ordinary photo.
In order to create an HDR photograph you need to have a post processing program like Photomatix or Photoshop, and ideally you want a tripod to ensure that your camera doesn't move between exposures.