HDR stands for High Dynamic Range, meaning quite simply, "a high level of contrast". HDR photography, therefore, is photography that captures a very high level of contrast, including very bright highlights that still have detail instead of being pure white, and very deep shadows that still have detail instead of being pure black. HDR photography is often captured by capturing multiple images at different exposures and blending them together, however many modern cameras can capture high dynamic range images in a single exposure.
Technical explanation of HDR Photography
When the term "HDR photography" was first coined, it only referred to a photographic technique which overcame the limitations of existing cameras: If you set your exposure so to preserve details in your highlights, many scenes were too contrasty and the shadows in that scene would be pure black with no detail. The same was true for exposing for your shadows: If you made sure to choose an exposure that obtained detail in the darkest shadows of a scene, the brightest parts of that scene would most surely "blown out", or pure white.
This shortcoming was defeated by bracketing the camera's exposure, usually with three or more images captured; one that fully preserved highlight detail, one that fully preserved shadow detail, and one or more somewhere in between, just for good measure.
These different exposures could be layered on top of each other in Photoshop, and blended together to create one single image that preserved details from the brightest highlight to the deepest shadow. The images had to be captured from a tripod, of course, in order to blend correctly. (And even then, any motion in the scene would still create errors in the blending process)
In recent years, however, modern digital cameras have increased the dynamic range capability of a single exposure by many, many EVs or stops. Thanks to various advancements in digital sensor technology, many digital cameras can capture fine quality details in both bright highlights and deep shadows with a single exposure.
Despite the lack of bracketed exposures and special HDR software, the resulting image from such a single exposure is often still called an HDR image, simply because of the level of detail it captures.
HDR photography went through an initial fad phase among photographers, and HDR processing was often highly unrealistic and the subject of much debate. Lately, however, HDR photography does not necessarily denote an over-processed image with unnatural looking highlights and shadows, it is simply the overall genre of photography in which high dynamic range is captured by any means, whether the results appear natural or "candied".