In high dynamic range photography, the scene being photographed is captured at multiple levels of light (exposures). This allows all portions of the scene, whether they are bright or dark, to be captured with sufficient detail in at least one of the different exposures. The brightest exposures provide detail in the darker areas of the scene, but wash out the brighter areas. Meanwhile, the darkest exposures provide detail in the brightest areas of the scene, but the darker areas are quite dim and hard to see.
Later during post-processing, the different exposures are combined using software to create a single high dynamic range image. The resulting HDR image includes all the details from across the scene, regardless of bright or dark areas particular portions of the scene were.
It is also common to remap the HDR image back into a standard dynamic range (SDR) image using a method known as tone mapping. Depending on the tone mapping method used, the resulting SDR image may look either realistic or unrealistic. Tone mapping is often necessary to share photos taken using HDR techniques since many of the most common image formats only support SDR.
More high dynamic range photographs I have taken can be found here.