HDR or High Dynamic Range photography is when you take at least three or more photos of the same subject bracketing the exposures by some amount and then using special software overlay the photos on top of each other. You then play with the tone mapping (color saturation, intensity, etc) to achieve different looks.
For the pictures above I shot three exposures with the over and under exposures being bracketed 3 f stops above and 3 f stops below from the proper exposure. I have a Canon 50D SLR so I can set that in the camera and rapidly shoot the three shots with minimal or no movement. Tripod is recommended. Many others will do the same bracketing or more.
When you bring the photos together using special HDR software such as Photomatix or FDR Tools, there are others. I used Photomatixs is that it overlays the over exposed parts of the image on to the under exposed parts and the under exposed parts on to the over exposed parts giving it a rich powerful look.
The idea behind all this is that the digital chip cannot see the full range of color as what the human eye sees. By taking multiple exposures and bringing them together you can more closely achieve what the eye sees. The fun part is taking it beyond that by playing with the tone mapping, etc and making the image more of a work of art than just rich in color and exposure.
That's my attempt of of explaining it. I've only been playing with it for a month or so.
Just do a search on HDR on the Internet and you'll find tons of information on the subject.
Anything else just let me know.