Dodging and burning is an old school technique that existed long before digital cameras and software. This does not mean it shouldn't be in a modern photographer's bag of tricks.
I am old enough to have learned dodge and burning in high school photography class. It was such a simple concept. The light coming out of the enlarger went through the film and projected your image onto light-sensitive paper. If you left it on too long or set it too bright, you'd end up with an overexposed, dark image. And if you were too dim or too fast, it would come out light or underexposed. Once you figured all that out, you would run into an image that needed more than just one setting. It needed parts of the image to be lighter or darker. Enter the dodge and burn technique.
Luckily for us all, there are no chemicals involved with today's type of dodging and burning covered in this Adobe Photoshop tutorial by photographer Michael Shainblum. All of the dodging and burning is done with a tool and your preferred input device. In the video, Shainblum uses and recommends a Wacom pen tablet.
Shainblum walks us through some of his decision-making and actual workflow techniques. The idea seems to be to enhance the existing highlights and refine the elements of interest in the photo. The video is a little longer, clocking in at 17 minutes, but the lesson goes beyond the technique itself and covers a lot of the why and reasoning behind the edits being made.