Uncaught Third Strike
If the catcher does not catch strike three, the batter can run.
How it actually works. The strikeout is recorded either way. What is not recorded is the out, until the catcher tags the batter or throws to first. That is why a pitcher can strike out four men in one inning.
Why it is done, and when it is wrong. The exception exists for the same reason the infield fly does: without it a catcher could drop a third strike on purpose with a runner on first and turn a double play. The rule closes the loophole by removing the option exactly where the double play would be there.
At Little League and travel ball. Little League Majors and below do not use it, so the batter is simply out. It arrives at the Junior level and in most travel ball, and it is one of the first rules that surprises a player moving up.
Batter's Eye: look up anything in baseball