HDR Photography stands for High Dynamic Range Photography, it is basically a heavy post-processing technique that allows you to take complete control over the tone of a picture which cannot be achieved by normal means.
Lets take an example, you take any normal picture (JPEG/GIF/PNG, etc..) and try to adjust the brightness or contrast of the image using Photoshop or GIMP. You will soon notice that if one part of the picture gets the desired brightness, some other part of the picture will become very bright(overexposed) or very dark(underexposed), because of this you cannot modify the tone of any image to a great extent. HDR targets this problem and tries to give you better control over the tone of your image.
The idea behind HDR photography is to combine multiple pictures of a particular scene into a single image. Each of the source pictures are supposed to have different exposure settings, most of the HDR photos use 3 source images, one underexposed, one overexposed and one with normal exposure. When you combine multiple exposures into a single image, the post-processing software now has a range of exposure to work with (unlike a normal photo which has a fixed exposure value). This will allow the software to adjust the tone in a very different manner with complete control on every part of the image.
Let me take the following images
All the 3 images represents exactly the same scene.
Now you need a software like PhotoMatix (proprietary) or QtPfsGui (open-source) to chew these images and create a intermediate .HDR file. Now this HDR can be toned with various algorithms in the software and you can produce a variety of tones of the HDR which can be saved as a normal JPEG or PNG. One such tone that I achieved with the above images is
Just look at the details of the temple sculptures and at the same time observe the tone of the clouds, none of the source images can show both of them. When we combine multiple images, we merge the "best" part of every photo which yields a very pleasant image.
You can see lot of good HDR pictures here.