What does Adaptive Dynamic Range do to a photograph? How does it do its job? How does it affect the results? When should it be used?
Nikon calls this "Active D-Lighting"; discussion of other systems (and how they differ) would be appropriate.
I'm looking for specifics here, not just "make pictures look better in extreme lighting conditions." Nitty-gritty technical details might be appropriate too, but please be gentle.
Answer
The dynamic range captured by the camera is higher with more steps than can be stored in a JPEG image. Around 2^14 (there is loss due to noise, of course) versus 2^8.
The mapping of those values is known as a tone curve. Adaptive Dynamic Range, Highlight Correction, etc. usually underexpose the image slightly and apply a tone curve that everything to the correct exposure, while compressing the highlights and thus capturing more highlight detail. Many cameras already do this without any options on, if you look at dynamic range comparisons on DPReview, the non-linear highlights (all digital sensors capture light linearly), and DxO's ISO Sensitivity charts (cameras that do this have lower ISO sensitivity ratings)
Some DSLRs also have Shadow Correction, which doesn't underexpose the image, but just boosts the shadows as that is where most of the information is lost when saving to JPEG.
The result with Shadow Correction is a more noticeable contrast decrease, although you get more information to work with and you can always push it down afterwards. Highlight correction doesn't have a noticeable decrease in overal contrast, but rather just increases local contrast in the highlights and captures detail at the upper end of the exposure.
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