This rule applies when first base is open, or if there are two outs. If the batter strikes out (swinging or looking) and the catcher does not catch the pitch before it hits the ground, the batter can run to first base. If the batter fails to run, she can be called out when she enters the dugout.
When does it not apply?
If there are less than two outs and first base is occupied, a dropped third strike could lead to a sneaky double play. For example, with bases loaded, the catcher could drop the third strike, step on home plate to force out the runner from third base, and throw to first base for a double play.
What does it mean for players?
As the batter, players should break for first base on any third strike and run until their coach or the umpire lets them know the catcher did catch the ball. Don't wait and see! As a catcher, you should "catch, tag, or throw" with two strikes. A quick catcher will often be able to tag the batter before she starts running, which makes the play much easier and holds other runners. With two outs and bases loaded, the catcher only needs to step on home plate after a dropped third strike, forcing out the runner from third base.