"Dynamic range" refers to the range of luminance between the brightest and darkest parts of an image or scene. A film scanner, for example, might be praised for having a high dynamic range, meaning that it can register detail in both very high and very low density areas of a negative. Human sight is capable of processing a very high dynamic range, as demonstrated by the fact that I can sit in a room with a window and be able to see detail in both the couch in front of the window, and in the bushes outside the window. Most other recording and displaying media have much lower ranges. Negative film has a higher dynamic range than slide film, most digital cameras, and photo prints.
In traditional darkroom photography, the difference between the dynamic range of a scene and a negative can be dealt with by manipulating the exposure and development times to handle a greater range of light. The difference between the negative and the print can be dealt with by selectively exposing the print so that the darkest areas are exposed for less time than the lightest areas. This makes the dark areas a little lighter, and the light areas a little darker, compressing the total dynamic range from what it was on film to what can be captured in a print.
With digital photography, the sensor in a camera can't be manipulated to have a greater range than it is naturally built with. So in order to compress the dynamic range of a scene into that which can be printed, usually more than one exposure is taken, and those exposures are merged together selectively on the computer, much like selectively exposing a print in the darkroom. This can be done "by hand," which can be a very easy or tedious task, depending on the image. There are also programs and plug-ins that will take several exposures of the same image and merge them automatically. Both these programs and the images they create are generally called "HDR," which stands for "High Dynamic Range." The name is a bit of a misnomer, because they are actually turning scenes with high dynamic range into prints with lower dynamic range. The range is being compressed, rather than expanded.
Many of these HDR programs compress the dynamic range on a micro level, so that not only are broad areas of light or dark in a scene flattened out, but the individual highlights and shadows on every object in the scene are also flattened out. This process produces a look that often resembles video games and other computer generated graphics. That look has begun to be called "HDR," even when it wasn't created from a scene or set of files that had a high dynamic range to begin with.
So when someone takes a photo of a scene that has a medium dynamic range, and all areas of the scene are adequately exposed in the image, and then processes that image to look as if it had been put through an HDR program, and then calls that image an HDR image, it's wrong on multiple levels. It's a mislabel of a mislabel. The photo, first off, wasn't actually given any treatment to deal with a high dynamic range in the scene. And even if it had been, calling the final result a "high dynamic range" image is also a mislabel. The final image doesn't have a higher dynamic range, it has a lower one.
This is why I hate the term HDR.