dodge = make brighter
burn = make darker
These terms come from the film days, when you would use a "dodge tool" which is basically an opaque object on a wire, to block the light from the enlarger from exposing the print, thereby making that region lighter. Sometimes you could just use your hand. Burning is the opposite, you have a mask that you cut out and you hold it over the print. The distance away from the print determines the "diffusive-ness" of the action. The exposed area gets "burnt in" from the additional light, making the print darker in that location.
The decision of whether to dodge and/or burn had to be calculated alongside the total exposure time of the print and the nature of the image. For example, if a B/W negative was overexposed and too contrasty, you could try to recover the highlights by burning them, but depending on the image, what might work better for the overall tonality is to increase the print exposure time and simply dodge the darkest portions of the image to prevent those areas from becoming too dense in the print. In a sense, it is much more difficult to lose detail via negative overexposure because as dense as the film gets, it will never really block 100% of the light during printing. You'll get a lot of highlight compression and it won't look as good as a correct exposure, but there is at least some information still present, whereas underexposure is a disaster because you've failed to capture the information in the first place. Once the film is developed, it is essentially clear.
Some people look at the complexities of digital photography today and long for a return to film, all the while forgetting that film was just as complex, with all kinds of issues regarding exposure, development, and printing, and its own specialized language and nomenclature. It was also expensive--a respectable professional would have their own darkroom, enlargers, and chemistry. Digital has brought high-quality imaging to the masses, but it is no easier or harder than film for those who seek how to obtain best possible output.