The National League had a three-way race: Cubs, Giants, and Pirates (led by Honus Wagner). The Cubs got their break when, in a game against the Giants, one of the Giant's rookies named Fred Merkle made a rookie mistake, which enabled the Cub's Johnny Evers (you know, Evers to Tinkers to Chance) to take advantage of a fan melee on the field causing the Cubs to win the "Merkle Game", which created a tie between the Giants and Cubs at the end of the season, which cost the Giants the pennant when they replayed the Merkle Game and lost to the Cubs.
From How Stuff Works:
The score was tied 1-1 and the sun was setting over the Polo Grounds in New York. Fred Merkle, a rookie substitute, was standing on first and Moose McCormick occupied third with two outs in the bottom of the ninth, when Giants shortstop Al Bridwell singled to center.
Thinking the game was won, and with a crowd of happy fans swarming the infield, Merkle bypassed second base and made for the New York clubhouse. But Chicago second baseman Johnny Evers got the attention of the umpire who, after seeing Evers tag second base with a ball (there was some dispute over whether it was actually the game ball), declared Merkle forced out at second, nullifying the winning run.
The two tight pennant races would have been enough to guarantee the 1908 season a special place in history, let alone the Merkle Game. But the season was also full of dirty baseball (not just by Ty Cobb, but he certainly helped), double-dealings, and an attack by a swarm of gnats (just like the attack on game two of the 2007 American League Division Series featuring the Yankees and Indians).
I learned all I know about the 1908 season from Crazy '08 by Cait Murphy.
Here's a reprint of a Keith Olbermann article about the Merkle Game. Olbermann writes about the game every year in hopes to clear Merkle's reputation.
Here's another article about it.