They are ways to adjust exposure in certain areas.
Burning is when you give an area extra exposure to darken it. In the darkroom, you would typically use a piece of black cardboard with a hole in it, then shine the light just onto the area where it is needed.
Dodging is when you give an area less exposure to lighten it. In the darkroom, you might use a thin stick to hold a small chunk of paper to block out the area.
In both instances you want to make sure to move it around a bit, so you don't get deep exposure lines.
I hope that makes sense...