Officials - Umpires and Scorers
As with any game, there have to be officials to manage the game, and decide on contentious issues. In Baseball, the officials are called Umpires.
Normally Umpires work in a crew of four, taking it in turns to occupy the role of "Home Plate Umpire" for that day's game. The home plate umpire is the most important official in any game. He stands just behind the catcher, and decides whether a pitch is a strike or a ball.
It is generally acknowledged that whilst the umpires are all supposed to call the same strike zone, each umpire actually has his own "individual strike zone". This isn't (usually!) an issue of contention, all that pitchers and managers ask is that the umpire calls the same zone for both teams - if he isn't consistent with that during a game, then you can all but guarantee that the manager will protest from the dugout. He's not allowed to come out and argue balls and strikes with the umpire, though this won't always stop him. If he does he'll be ejected, but it's generally considered better that the manager gets thrown out of the game for shouting at an umpire than his pitcher does!
The home plate umpire wears a mask (the same as the catcher) and a chest protector (the same as the catcher) and other items of protection. He relies on the catcher either snaring balls that get past, or getting in the way and being hit first! You can usually spot that the home plate umpire has a bag of fresh balls hanging from his waist, to replace any scuffed ball with a new one.
The home plate umpire will normally call a "strike" by throwing his arm out in a sharp manner, or clenching his fist in an upward motion, whilst a "ball" is called without any signal (if he doesn't make a signal, it's a ball). His call of a third strike will usually be very exaggerated.
In addition to the home plate umpire, there will also be a "first base umpire", a "second base umpire" and a "third base umpire", each of whose job is to make decisions about whether runners are "safe" or "out" at their particular base. "Safe" is normally called by a sweeping motion of both arms whilst "out" is called by pumping one arm as if trying to beat a door open.
The first and third base umpires are also responsible for calling whether balls hit along their lines are "foul" or "fair", though in the playoffs two other umpires are added to stand in the outfield along the first and third base lines to make outfield calls.
Finally, the first or third base umpire may be called on by the home plate umpire to advise whether a batter "went around on a check swing". If a hitter starts swinging at a pitch, and then realises it's going to be a ball, and checks his swing, then it's only a strike if he swings his bat in front of him. If he manages to stop the bat before it gets in front, the swing doesn't count.
The umpires will (usually) work as a team, and if any umpire has any doubt about a call they'll congregate to discuss the call and make sure they get it right.
The Official Scorer/Fielding Errors
The Official Scorer is not an umpire, and has no part in determining the course of a game, but some of his decisions may be key for the players concerned, particularly when contract time comes around!
In Baseball, a key statistic is the "Error" (or "E"), which is defined as "a misplay (fumble, muff or wild throw) which prolongs the time at bat of a hitter or which prolonges the life of a runner, or which permits a runner to advance one or more bases". The Official Scorer is given the guideline that "this must assume an ordinary effort by the fielder".
In a nutshell, if a fielder makes a mistake which allows a runner to advance, or prevents him from being out, then he's "charged with an error". The pitcher is then usually not "held responsible" for that hitter/runner - "he's already done enough to get him out once, so why should he be penalised for someone else's mistake?" is the logic.
The main statistic still used for measuring the ability of a fielder is still the error, a player's "fielding percentage" (the proportion of times he handles the ball without making an error) is often quoted. It's ironic that no-one ever seems to have invented a reciprocal statistic - the "field save". If an error is a fielder messing up and turning an "ordinary out" into a base hit, why not have an equally subjective log of a fielder turning an "ordinary base hit" into an "extraordinary out"?