A successful dodge in Pokemon Go avoids 75% of the attack's damage. The gif below shows users how to dodge:
Swipe left/right within 0.7 seconds after the "yellow flash".
Ignore everything else about the defender's attack animation, just swipe after the yellow flash. Some attacks, like Blizzard, will appear to hit you before the yellow flash ever appears, but follow this rule and you will always avoid 75% of the damage.
Dodging small attacks probably isn't worth the time cost. Most special attacks cause the defender to pause and sit still for an extended time before the attack. Use this indicator to know a special is coming and be ready to swipe immediately after the yellow flash.
Queuing up moves by spamming makes it difficult to dodge.