The simple answer is this.
HDR is an acronym for High Dynamic Range
What is Dynamic Range?
It refers to how we see light and what we see. Its measured in stops of light, our eyes see about 13 – 14 stops of light, most consumer digital cameras will record about 9 stops of light.
We can see details in the bright and dark areas of a scene that our cameras cannot record. In some cases it is the answer to “Why don’t my pictures look like what I was seeing”.
This photographic technique allows you to get more detail into your pictures than your camera is capable of recording by taking multiple shots that are set to different exposures and combining them together.
Usually three or more shots are taken: underexposed, correctly exposed and over exposed (-,0,+). Your camera may be able to do this automatically with a function called “bracketing” – your manual may explain this in further detail. These multiple images are then blended with third party software or in camera to create one image.
This technique is handy when a scene is unevenly lit, an example of this is a place called Natural Bridge, its in the Springbrook National Park in Queensland. There is a wonderful sun lit waterfall in a cave.
The shot below is exposed for the light on the waterfall and the light outside.
Note that there is no detail in the underside of the cave, but the rainforest and waterfall are exposed properly.
This next shot is exposed for the underside of the cave.
Note now that there is no detail in the rainforest outside and there is little detail in the wall and the waterfall.
The next shot is in-between both of the two shots above.
There is loss of details in the darker areas and loss of details in the rainforest, but there is nice light on the rocks in the foreground.
If we combine these images using HDR processing techniques, we will get the composite of the lights and darks. You can now see the details in the underside of the cave and the details in the rainforest.
This image was combined in Photoshop, using the built in HDR tools. Some of the newer cameras will process this style of image in the camera.
Some people believe that this is no longer photography, and is more akin to art. I believe that if you need to use this style or process to record what you see, then do it.