HDR means combining 3 or more bracketed exposures into a single photograph that has a dynamic range greater than is possible in a single exposure.
The "HDR look" that people are talking about is local contrast adjustment. Not all HDR images have this look, nor is HDR the only way to achieve the look.
Local contrast is comparing the lightness and darkness of a specific area of the image. As an example, take a BW photo of a face. If the lighting is soft and even, and maybe a little low, it wouldn't be uncommon for the whites of the eyes and the teeth to be almost the same gray tone as the skin of the face. If I go in and make the eyes and teeth as white as we imagine they are, it's way too much, and looks strange and unnatural. If I just lighten them just enough that they are a noticably lighter gray than the surrounding skin it will look more natural to the viewer, although if I use editing software to measure the tone, it may seem very dark from the tone I might imagine that teeth and eyes are.
Here are some examples of local contrast.
http://www.michaelbach.de/ot/lum_adelson_check_shadow/index.html
http://www.michaelbach.de/ot/lum_contrast-contrast/index.html
This is a HDR image created from seven exposures. I don't think it looks like what people have come to think of as HDR, because I didn't go crazy on the local contrast.
Here's a borderline example. The local contrast manipulations are more noticable in the small, online version than a full sized print.
This shot was taken in heavy shade. Exposing for the couple I'd blow out the far background and sky. Expose for the sky, and the couple are a silhouette.
The use of heavy local contrast manipulation may be faddish right now because tone mapping software makes it easy to do, and people get a kick out of it ( me too! ), but painters have been using it for centuries, and I'm sure photographers will figure it out how to use it effectively too. HDR as a tool is stunningly powerful. It means that as long as my subject is suitable to take at least 3 exposures of the exact same composition, I can go way beyond the normal dynamic range limitations of my film or sensor.
I agree that in many cases it is overused, or not used effectively. But then again 90% of everything tends to be crap, newfangled or not. Once upon a time color photography was a cheesy fad.