In photography, the acronym HDR means High Dynamic Range. A High Dynamic Range is the measure of the lightest tones (highlights) to the darkest tones (shadows).
The human eye is capable of seeing a wide dynamic range (i.e., in photography terms, a range of 11 or more "f-stops"), which is why we can see details in shadows as well as details in highlights. Whereas, cameras have a shorter dynamic range (depending on other variables, a range of 3-5 "f-stops"), meaning they catch fewer details at the high and lower ends of contrast.
High Dynamic Range photography is the method of merging images to create a final composite photo. The method involves taking many pictures and stitching them together in a composite photograph. This method can give the photograph a dramatic effect.
To accomplish this, a photographer may take a series of photographs (normally at least 3) at different exposures values. Each photograph captures the image at a different exposure. Typically this will be one normal exposure, one over exposed, and one under exposed. For instance, one can use bracketed shots at -2, 0, and +2.
Most of the DSLR cameras offer an automated way of performing these image capture using an automated feature called bracketing. But this can also be done on cameras that allow for manual settings or the ability to over-ride the exposure settings.