It's astounding that dodge and burn didn't originate as first person shooter lingo for strafing with a flamethrower, much less something coined with the advent digital image editing. The truth - it's as old as photo development. In the darkroom, burning is the process of letting something get more light and thus become darker, and dodging blocks light and leaves it lighter. Simple. Logical. Powerful.
In Lightroom this is a piece of cake, and here's an example. This was an excellent photo, but Leah's head was directly under an overhead light leaving her overexposed, while I was shielded from the light and rather shaded.
I was able burn her head in (sounds terrible) and get back some of the highlights, then dodge some on my head and shoulder to get a little detail. I actually like shadows as they are away from the action, but this let me recover a great image.
For perspective, I was just reading about people who looked at negatives from some of Ansel Adam's greatest photos and found that they were relatively unimpressive if you made a straight print. What this means is that his amazing images were to a significant extent creations made in the darkroom, and dodging and burning were the techniques he used to brighten and darken parts of the landscape to create something greater than what was directly captured on film. Decades later and nothing has changed.