Cutoffs and Relays
A fielder lines up between the ball and the base so a long throw becomes two short ones.
How it actually works. On a ball to left or center with a throw going home the third baseman lines up the cut and the pitcher backs up the plate. On a throw home from right the first baseman cuts. On a ball into a gap or a corner the nearest middle infielder goes out as the relay man and the other one trails him a few steps behind, so that a wild throw has somebody standing behind it. The cut man's job is to be exactly on the line between the fielder and the base, at a distance where he can still make a strong throw of his own.
Why it is done, and when it is wrong. The cutoff man is not there to shorten the throw. He is there to change the target after the throw is in the air, which is the only way a defense gets new information into a play already in motion. The catcher makes that call because he is the only person facing the field, and he is choosing between an out at the plate and everything else that is happening behind it. That is the real answer to when you throw to second instead of home: you do it the moment the runner heading home is no longer catchable, because the trail runner reaching second turns a run into a run plus a runner in scoring position, and that second runner is usually the one that beats you.
What people get wrong about it. The most expensive throw in baseball is a strong accurate throw home that arrives two steps late while the batter takes second unchallenged. It looks like effort and it is a mistake. The instruction to hit the cutoff man exists precisely because outfielders want the highlight and the throw that skips past everybody costs a base every time it fails.
At Little League and travel ball. Youth outfielders throw home on everything, and it is the largest single source of extra bases against a youth defense. The fix is a rule rather than a judgment: at these ages the ball goes to the cut man unless the coach has said otherwise before the pitch.
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