I grew up in Maryland, as a Red Sox fan. It was in my blood, it came down from being born in the Fenway district, it came down from my grandmother, perhaps, who I remember did follow the Sox. It was hard being a Sox fan in Maryland at the time; these were the dark days after 1986, mostly the early 90's in my memory, and all the news for baseball that I heard was the Orioles, and specifically Cal Ripken, Jr. The Orioles, I seem to recall, were doing fairly well in those years, they and Toronto, while the Sox and the damned Yankees languished in some sort of netherland - and with neither the Sox to cheer for or the Yankees to protest against... I checked the standings every so often, tracking games behind, calculating rough guesses about numbers of games left (this was before I learned enough to play with magic numbers and well before I thought it worthwhile to follow the careers of particular players with any interest).
I got A's trivia, analyses of Jose Canseco (who amused me immensely by appearing in the Sox roster, as I actually knew his name), wavings about of baseball cards and half-hearted parsings of statistics from my brother. From National Public Radio, I got the labor disputes and discontent, and The Streak. From The Washington Post, I got Cal Ripken - The Streak again. The wonder of having a shortstop who could hit.
And I wondered as I listened, what was so wonderful about a shortstop, in particular, who could hit? Was there something inherent in the position that led, difficult though it may be to believe granted Nomar and Jeter, to less than optimal hitting skill? It was one of those mysteries that I couldn't manage to resolve. Why can't shortstops hit? Why are first basemen sluggers?
Catchers I could figure out fairly easily. The typical catcher is a big, solidly built guy who needs to be able to take the abuse and strain of the position, the occasional wild pitch, mislaid bat, or runner desperately trying to knock him over and claim the plate. He has to be able to make the sharp peg to second with enough strength to catch a fast runner trying to steal. The catcher is not known for speed, but rather for power; the strength that makes for the stubborn wall guarding the plate is not necessarily one that translates well into the finesse of hitting for average.
That didn't help me figure out how the other position stereotypes were established. I asked the question of the Bosox mailing list, and got some reference material and a bit of explanation that helped me figure out more or less what the actual answer was, in a statistical sense. I've puzzled out some less esoteric reasonings for the positions I didn't get significant input on, rambling somewhat but going for logic.
Primary among these references was Bill James' defensive spectrum; this is something which was developed over years of sabermetric analysis. The principles of the defensive spectrum are, in base, that it is possible to move from 'right' to 'left' over the course of a career far more easily than the other direction, because the leftward positions are easier to play, and that it is much more difficult to find a good player for the rightward part of the spectrum than the left, for much the same reason. The spectrum is:
Thus, shortstops who can hit are 'rarer' than other positions because a good shortstop - in general - is hard to find. And, Buckner aside, it's much more difficult to make errors at first base than other positions; the first baseman mostly has to field and guard the bag in the event of pick-offs - more errors are throwing errors than pure fielding errors. So if one doesn't need someone who's particularly brilliant at the position defensively, it's best to put a slugger at first.
Second basemen are stereotypically quick (base stealers, among other things) because they need to be able to turn the double play; the middle infielders get a lot to do. According to Yahoo's "All Average Team", second basemen tend to be good average hitters and score runs as well - which all in all explains why Offerman's as good a leadoff man as he is. He gets on base, he advances on the bases, and he gets home quickly when he has a chance.
Center field is a more difficult realm to play than the corner fields, of course, because of the amorphous boundaries of the fields; CF has to be aware of both sides, and be prepared to back up the middle infielders (though mostly 2b, since LF backs up SS) or either of the other outfielders if the ball is headed into an indistinct zone. The entire middle zone - Catcher, Shortstop, Second Base, and Center Field - is the most critical to baseball fielding defense.
The hot corner is not as easy to play at first, and needs a guy with a good arm - so as to have a chance of getting the out at first on a 5-3 play. However, with a good shortstop, 3b mostly worries about fielding line drives (hard to catch under the best of circumstances when they have height and often bound for the outfield), pop-ups (easy catches), and down-the-baseline dribblers, and keeping runners on third from heading home by blocking some view of the outfield and being ready to field pickoffs. However, it's certainly easier to handle than the middle infield.
The corner outfield positions are traditionally associated with heavy hitters - Ted Williams being a fine example of that tradition. Most balls that reach the outfield are relatively easy to catch flies or the more difficult long-driven liners. The more difficult fielding calls - the ones that carom off the walls or are near-homers - are relatively rare, and in many ways are the most park-dependent of all hits, giving the home outfielder a significant advantage. A ball played off the Monster is a different skill than a ball played off the corner in Yankee Stadium or Camden Yards.
I'm not certain how the spectrum on RF/LF was determined logically, though I'm certain there's statistical backup for the order in the spectrum. I believe that left field gets a slightly higher proportion of long flies, granted the preponderance of right-handed batters, and the tendency of right-handers to pull to the left (I always pulled left on hits; I usually fouled right, though - not that I'm anywhere near Major League quality in my batting). However, right field is closer to first, and thus a non-fly ball hit there seems more likely to be outs with a good right fielder; the ball's more likely to get to first before the runner for the out with the shorter distance. A right fielder therefore has to make the decision about whether to throw for first for the attempt at the out, or to second to be sure to hold the runner, much more quickly than left field - which has a longer, more difficult throw to first and might well throw to second to hold the runner to a single instead of going for the out.
And then first basemen, who, since they don't need to do as much of the throwing part of fielding, might as well be offensive powerhouses.