Agree 100 percent with DTaylor's comments.
Adding a very biased personal perspective.
In the film days, this was called the Zone System and it was popularized by Ansel Adams. The basic concept remains the same. Film (now sensors) and the final medium (photographic prints, photo-mechanical reproduction and now, monitors) cannot reproduce the full range of tones that exist in nature.
You have two choices when trying to capture scenes with a broad range of tones: You can "clip" the highlights/shadows by letting them go either completely white (for highlights) or completely black (for shadows). Or, you can try to compress the tonal range into something that is reproducible, preserving detail in both.
Right now, the fad in photography is to try to compress as much of the range as possible, creating "High Dynamic Range" scenes, which is a misnomer, because the range is not actually "higher" instead it is compressed and often looks very artificially so, like the example shown early in this thread. Some people think that looks great. Others think it is artificial looking and excessively dramatic.
Ansel Adams was a big proponent of compression and his Zone System was built entirely on teaching people how to compress the tonal range of film and paper. However, Adams' goal was to try to reproduce a tonal range that looked natural and reflected how the human eye (at least in his view) actually saw the scene. Many of today's HDR images make no attempt to reproduce the scene as a human eye might see it, but instead opt for overly dramatic misinterpretations.
The counterpoint in photography is the image that allows areas to go pure white and pure black while retaining the important detail in the middle tones.
As with most things in life, the decision should be personal and should be based on what your goals are. In my view, moderation is probably the best approach. But, that's a subjective decision.
The appeal of a wider dynamic range in sensors is that the sensor can preserve shadow and highlight detail in the file that can be brought out in the final image. Again, as DTaylor has pointed out, this can often be done in post-processing in Photoshop. I frequently use multiple layers of smart objects, adjusting the highlights and shadows in Camera Raw, masking certain areas and combining them in Photoshop. But then, I'm not a big fan of the HDR look and in most cases prefer the challenge of composing and capturing scenes within the limitations of the camera's existing dynamic range.