HDR stands for high dynamic range and it's a technique that merges three different exposures in one image. Why would you want to do that? Because sometimes the light is far from ideal. Take a cloudy day, for example. The sky will have a fantastic texture given by the white layers of clouds, but in order to capture it you have to use a short exposure. However, when you do that, your foreground will be too dark. The human eye resolves that because as you shift your eyes from one focal point to the next, your pupil contracts and relaxes, allowing more or less light to get through. Your camera can't do that. By taking two or three images of which one should be under-exposed, and one over-exposed, HDR will allow you to take the best out of each shot.
I'm still struggling to use it in a proper way, but these are a couple of examples I took this past summer:
The original shots for the first photo are here, and here are the ones for the second. As you can see, none of the original shots is ideal, but when combined together through HDR, the end result is quite pleasant. Both pictures were taken in Ogunquit, Maine. It was June, and the weather was often gloomy, so I ended up using HDR a lot. I wish I could use it better, but I'm still learning. Also, notice that none of these examples were taken with a tripod. Ideally, you really want a tripod to make sure the three images you take overlap perfectly. However, since carrying a tripod around happens to be a little inconvenient, I noticed that shooting architectural elements helps because the software has an easier time overlapping the images along straight lines. Finally, the softer colors compared to the originals are due to a post-HDR processing tool called tonemapping.
Besides tonemapping, HDR comes with a whole suite of editing options, and depending on which you pick, you might end with pretty dramatic effects, like this:
(Original shots here.) Slightly reminiscent of Armageddon. Well, the light was a little dull, and my mood reflected the spirit, so I kept it. But, just so you know, the example above is NOT good HDR, IMHO. In fact, let's not call it HDR. Let's call it a personal interpretation, or an artistic rendition of an image. What usually makes HIDR-processed images look surreal and dramatic is an exaggerated use of tonemapping, fusion, compression, and all those fancy processing options. In fact, some of those options like tonemapping are available even without HDR. All you're doing is changing the color map on the image. But a well-dosed, expertly used HDR does not make the image surreal. It should only blend the light exposure.
Unfortunately, unless you're willing to spend a fair amount of time learning the technique, the "dramatic" kind of HDR is what you'll end up doing for the most part. And that's why so many people feel strongly against HDR. They claim it's artificial and doesn't reflect reality. People who take that stand should be aware that they are not criticizing HDR. What they find unreal and artificial is the software that manipulates the image after the HDR has been done. HDR per se is a fantastic tool, and like with all tools, it takes time and dedication to learn how to properly use it.
Here's what a professional photographer can do with it. (I've been following Jeff Sullivan on Flickr and on Panoramio and his photos are absolutely stunning. I'm hoping that by staring at his amazing pictures some of his talent will eventually rub off... well, I gotta hope for something!) National Geographic also has a nice HDR gallery (and some of those example I do find a little dramatic, but I admit they are very beautiful and well done).
So the bottom line is: photography is an art, and thanks to modern technology and software there are many, many new tools out there. Each tool gives you a little something and takes away a little something, and you have to pick and choose what you want/need/etc. Document yourself and rather than nixing one completely and embracing another one completely, and keep an open mind on all because you never know what you're going to love next.