The Manager and the Coaching Staff
The man in uniform who makes the substitutions, plus a staff of a dozen that most fans never see.
What it actually is. The manager runs the game: lineup, pitching changes, substitutions, defensive positioning and challenges. He wears a uniform because he is allowed on the field. Around him is a staff, typically a bench coach, pitching and hitting coaches with assistants, a first and third base coach and increasingly a run prevention or catching coordinator.
What somebody who knows the game gets out of it. The manager's in game decisions matter less than people think and his roster and clubhouse management matter more. Bullpen usage, rest, and lineup construction are worth a few wins a year in total, and analytics departments now supply most of the inputs. The third base coach, by contrast, makes a high leverage decision several times a game with about a second and a half to make it, and a coach who is too aggressive on sends can cost more runs over a season than the manager's tactical choices gain.
The numbers worth knowing. A manager makes perhaps three or four close decisions a game. The gap between a good and a bad third base coach on sends is measured in a handful of runs a season, which sounds small and is comparable to what the entire bench contributes.
What people get wrong about it. The manager does not call pitches in most organizations, and increasingly does not set the lineup alone either. What he does own is the pitching change, which is the single decision with the largest measurable effect, and the one fans second guess most.
At Little League and travel ball. At youth level the coach's decisions matter enormously more, because pitch counts, batting order and where players are positioned have a much larger effect on a six inning game with a run limit. The youth version of the third base coach problem is also the same: sending a runner at 10U succeeds or fails on the outfielder's arm, and most coaches under estimate it.
Batter's Eye: look up anything in baseball