High Dynamic Range imaging is the process of increasing the range of brightness found in a scene.
When you take a photo, that photo is taken at a certain exposure level. If you were to take two photos, one after another, one with a higher ISO and one with a lower ISO, everything in the photo with the higher ISO would be brighter. The same is true if one photo were taken with a larger aperture, or if one photo were taken with a slower shutter - these photos would be brighter in every aspect.
However, what if there's a very bright spot and a very dark spot in our photo? Would we rather the bright spot turn out dimmer, or would we rather the dark spot turn out brighter?
HDR eliminates this compromise. When you capture an HDR photo, your camera captures multiple photos as fast as it can, each at different exposure levels (one brighter, one darker, etc.). Your camera's processor then merges these multiple images into one image, with the dark spots showing as a deep dark and the brightest areas remaining a vivid, bright color.
Simply put, an HRD photo is one where the range in brightness is high.