Fielder's Choice
The batter is safe because the defense chose to get somebody else, and it counts as an out against him anyway.
What it actually is. A fielder's choice is when a batted ball is fielded and the defense throws out a runner ahead of the batter instead of the batter. The hitter reaches base but is charged with an at bat and no hit, so his batting average drops exactly as if he had made an out. It is scored that way because in the scorer's judgment he would have been out had the fielder chosen him.
What somebody who knows the game gets out of it. This is the rule that makes batting average a worse measure than it looks, because it is the one place where a hitter is punished for something a fielder decided. It also explains a real strategic idea: a fast hitter with a man on first changes what the defense can do, since a shortstop who has to hurry the pivot may get neither man. Speed at the plate is worth something on ground balls even when it never shows up as a stolen base.
The numbers worth knowing. A team is involved in a few hundred fielder's choice plays a season. They are common enough that a hitter with a lot of them and a high ground ball rate has a meaningfully depressed batting average.
What people get wrong about it. Reaching on a fielder's choice, reaching on an error and being safe on a dropped ball look identical from the stands and are three different things in the book. On base percentage counts none of them. This is why a hitter's times on base and his OBP can disagree with what somebody watching remembers.
At Little League and travel ball. At youth level the fielder's choice is where the real coaching lives, because the wrong choice is the norm. A 10U shortstop who reflexively goes to second with a slow runner at first and a fast one at the plate frequently gets nobody. The rule of thumb worth drilling is take the sure out, and Read the Play on this site is built around exactly those decisions.
Batter's Eye: look up anything in baseball