As it applies to traditional darkroom processing of prints, dodging means that part of the exposed image is shielded from some of the enlarger's light, the result being that the portion of the print being exposed comes out lighter.
"Burning in" means that a little bit of EXTRA light is allowed to reach the print, usually by shielding the rest of the exposure. The result is that the part burned in comes out darker on the print.
Traditionally, the tools used for this were a circular piece of something, like carboard, mounted on a thin wire, or, a hole cut in a piece of paper. Sometimes the photographer's hands and fingers could do the job.
These are methods used to make small corrections to a print when the negative is not quite perfect.
The digital equivilant to these steps, of course, eliminates the negative, but the results are the same. A "dodged" section comes out lighter, and a "burned in" section comes out darker.