HDR (High Dynamic Range) can be called a technique where you take multiple shots at multiple exposures and combine them to get a perfectly exposed picture with best possible details. Its best applicable where the scene contrast is the most. For example, if you're taking a picture of the sky (bright) through the window of a dark room (dark), you have two options:

1. You keep the sky perfectly exposed (with clouds and other details) but make the room/window underexposed (no details, almost black).
2. You keep the window/room perfectly exposed (with textures and other details) but make the sky overexposed (no details, almost white).

So, the workaround is, you turn on exposure bracketing (-2, 0, +2) of your camera, set the metering to evaluative and take 3 shots (continuous mode, self timer 2 second) of the scene from some steady place (tripod if possible).

1. The first shot should be overall underexposed and will keep the sky perfectly exposed with all details.
2. The second shot should be overall balanced in exposure.
3. The third shot should be overall overexposed and thus the details of the window/room will be there.

Now if you use Photomatix or any similar software to merge these 3 shots into one, with minor adjustments you can get detailed sky as well as details window/room as well.

Example:

Here you can take a look at some amazing HDR photographs. http://www.stuckincustoms.com/hdr-photography/

Also, this is a great tutorial for the beginners: http://www.cambridgeincolour.com/tutorials/high-dynamic-range.htm