HDR, or High Dynamic Range, is a technique that merges 2-7 images, taken consecutively, each which a different exposure. This is accomplished by bracketing your frames, setting the Exposure Compensation/AEB Setting on your camera to whichever 2-7 exposures you desire and then shooting in continuous mode. The blending process in post-processing extracts the best pixels in each shot to make a dynamic photograph which can be adjusted for a subtle or dramatic affect.
Below is an example of three shots I took recently... the first three are the bracketed shots and the 4th is the end result:
HDR certainly has its critics. Lots of photographers run the post-processing gamut from realism with slight enhancement to over-saturation that looks cartoonish. Keeping in mind that art is subjective, so neither is wrong.
There are two main reasons, in my opinion, why HDR should be used in certain circumstances. 1- it can save an unsaveable scene that is doomed by shadows and lack of detail too dark to make for a good image or washed out backgrounds. 2- to take an ordinary scene that that may be perfectly fine and transform it into something dreamy. An ordinary shot of a bench overlooking the water may lack any real appeal... ("That's a nice shot... but why would I want to hang it in my home?") but can be transformed into something extraordinary, conveying what the photography sees in her heart when she captures a single shot.
Trey Ratcliff of Stuck in Customs is regarded as the master of HDR, and rightly so. One of his HDR shots was the first ever to be hung in the Smithsonian Museum in Washington, DC, and he has propelled the craft of HDR to a respectable art form. His images are incredible. Anyone interested in tutorials on HDR should really check out his site.
You don't have to get it right every time with HDR. That's part of the beauty of photography. If you choose to shoot literally or with a dose of HDR, your art is your art, and you won't please everyone so concentrate on honing your craft and making God happy with what you produce. You can't do much more than that.