To dodge something is to avoid it. In dodge ball, players dodge the balls being thrown at them.
Dodging is making quick, sudden movements, usually to avoid something. You have to move quickly to dodge a flower pot that's falling from a ledge. In football, running backs are good at dodging defensive players. Politicians do a lot of maneuvers and manipulations to dodge blame, and plenty of people dodge responsibility for something they're supposed to take care of. So any kind of evasion strategy is a dodge.