Everytime you move parallel to an enemy, if it shoots at you, it has a certain chance to target the tile where you were standing before you moved, instead of the tile where you are now, which represents you dodging the projectile. Dodgemaster (if I understand it correctly) increases this chance to 100%. For chainfiring enemies (former commandos, arachnotrons etc.) this will only apply to the first projectile in their attack, so the rest can still hit you.
This means that, for example, you should be able to move around a hell baron forever without him ever hitting you, as long as you don't get too close to walls and take splash damage. This is very useful when fighting the Cyberdemon, since you can move around until he fires and then get free shots at him. Once you hear him reload, you should move around again until he shoots etc.
His rockets have a large splash damage radius, though, so you have to be careful where you dodge, ideally nowhere close to the pillars.
Edit:
This reminds me that I wanted to ask - just how useful dodgemaster is when you have tactical boots/armor and 2 levels of hellrunner? All of these increase the chance to dodge, and it should probably be around the normal max of 95%. Is the guarantee that you won't be hit by 1 in 20 shots worth taking the trait, or is it sometimes better to take another hellrunner for speed or any other trait?