Lead-Off
The old-school book says to put a speedy guy up top. Power isn't important, and OBP is nice, but comes second to speed.
The Book says OBP is king. The lead-off hitter comes to bat only 36% of the time with a runner on base, versus 44% of the time for the next lowest spot in the lineup, so why waste homeruns? The lead-off hitter also comes to the plate the most times per game, so why give away outs? As for speed, stealing bases is most valuable in front of singles hitters, and since the top of the order is going to be full of power hitters, they're not as important. The lead-off hitter is one of the best three hitters on the team, the guy without homerun power. Speed is nice, as this batter will have plenty of chances to run the bases with good hitters behind him.
The Two Hole
The old-school book says to put a bat-control guy here. Not a great hitter, but someone who can move the lead-off hitter over for one of the next two hitters to drive in.
The Books says the #2 hitter comes to bat in situations about as important as the #3 hitter, but more often. That means the #2 hitter should be better than the #3 guy, and one of the best three hitters overall. And since he bats with the bases empty more often than the hitters behind him, he should be a high-OBP player. Doesn't sound like someone who should be sacrificing, does it?
The Third Spot
The old-school book says to put your best high-average hitter here. The lead-off hitter should already be in scoring position and a hit drives him in. Wham, bam, thank you ma'am.
The Book says the #3 hitter comes to the plate with, on average, fewer runners on base than the #4 or #5 hitters. So why focus on putting a guy who can knock in runs in the #3 spot, when the two spots after him can benefit from it more? Surprisingly, because he comes to bat so often with two outs and no runners on base, the #3 hitter isn't nearly as important as we think. This is a spot to fill after more important spots are taken care of.
Cleanup
The old-school book says to put your big power bat here, probably a guy with a low batting average, who will hit the big multi-run homeruns.
The Book says the #4 hitter comes to bat in the most important situations out of all nine spots, but is equal in importance to the #2 hole once you consider the #2 guy receives more plate appearances. The cleanup hitter is the best hitter on the team with power.
The Number Five Guy
The old-school book says the number five guy is a wannabe cleanup hitter.
The Book says the #5 guy can provide more value than the #3 guy with singles, doubles, triples, and walks, and avoiding outs, although the #3 guy holds an advantage with homeruns. After positions #1, #2, and #4 are filled, put your next best hitter here, unless he lives and dies with the long ball.
Spots Six Through Nine
The old-school book says the rest of the lineup should be written in based on decreasing talent. Hitting ninth is an insult.
The Book basically agrees, with a caveat. Stolen bases are most valuable ahead of high-contact singles hitters, who are more likely to hit at the bottom of the lineup. So a base-stealing threat who doesn't deserve a spot higher in the lineup is optimized in the #6 hole, followed by the singles hitters.