Since we've gotten this far down the thread with nobody answering the question directly...
HDR stands for "high dynamic range".
The human eye can actually perceive a far larger range of dark-to-light than a camera can record or a computer monitor can represent. It's roughly twice as much--the human eye can see about 10EV, while the camera/monitor can do about 5EV (EV [exposure value] = stop).
It's always possible that a scene may contain more dark-to-light than can be captured in a single exposure. Think of a shot from inside a house through a bright window. If you expose for the scene through the window, the inside of the house will simply be a dark silhouette. If you expose for the room, the scene out the window will be completely blown to white. This is also why you can get white skies in shots.
HDR is one method for combining multiple bracketed exposures to cover the full dynamic range in a scene. The dynamic range is represented by larger numbers than are standard in most graphic formats, so the combining of shots with HDR also requires a different file format. This format cannot be viewed without special HDR displays. So if you want to make it viewable on a computer, you then need to "tonemap" the HDR file to a standard graphic file format, like JPEG or TIFF. Tonemapping maps one of the HDR color tones to a color in the visible range--you're basically squeezing everything back down to a smaller dynamic range. It's similar to the "fill light" adjustment in Lightroom.
Because you have so much extra information, you can also push things like saturation and that's how you get the heavily-processed HDR look. But you can also map out to make a scene look natural and just represent the full tonal range. It depends on how you choose to tonemap.
There are other methods to handle a high-dynamic range. In camera, you can use a graduated ND filter to darken skies. Or you can use multiple exposures, masks and layers to combine the sky of one shot with the landscape of another. Or you can use something like
enfuse, which automatically combines the exposures to a finished image without requiring going to an HDR file format or tonemapping back out.
But if you want the "exploding sky" effect, your best bet is probably
Photomatix. My personal druthers are the more natural-looking methods of using HDR, but different squids for different kids.