The World Series (and the Playoffs)
At the end of the regular season the three divisional winners (ranked 1, 2 and 3 according to their win-loss records) and two wild-card teams (ranked 4 and 5) from each league (the American League and the National League) go forward into the playoffs.
The two wild-card teams play each other in a one game "winner takes all" match, with the survivor joining the three divisional winners in the "Divisional Series" (effectively the quarter finals, as there are two pairs of teams in each league). The four teams in each league play two series of knockout baseball to determine the who "wins the pennant" (by tradition, the league winners are awarded a pennant to fly over their stadium). The two league winners then meet in the "World Series" (believed to be so called, because the first sponsor was the "Daily World" newspaper).
The first round of the playoffs (known as the "Divisional Series") are played over the best-of-five games (the first two at one venue, then two at the other, then a fifth at the first venue if required), with the higher ranked team having "home field advantage" (i.e. three home games and two on the road). The second round of the playoffs (known as the "Championship Series") are played over the best-of-seven games, with home field advantage meaning four home games (two games at one venue, three at the other, then two more at the first venue).
In the first round, the no.1 ranked team plays the no.4 ranked team (and no.2 plays no.3), unless they came from the same division, in which case 1 plays 3, and 2 plays 4. Ties in rankings are broken by the toss of a coin!
Note: the Major Leagues keep changing the formats for seedings, who plays who and schedules, and the TV companies have an impact on scheduling as well.
If a team can clinch a postseason series early it's a big advantage, as they can rest their players and "set up" their pitching rotation for the next series (if your star pitcher had to play in the final game of the Divisional Series, he won't be rested enough to pitch until the middle of the Championship series). However, if a hitter doesn't play for more than a day or two he tends to get out of rhythm, and teams with a few days off will normally play one or two practice matches (but not involving their pitchers).
Normally (unlike the regular season) there is a day's travel in the middle of the series when the venue changes, so teams can normally have smaller pitching rotations for playoff series (because of the extra days' rest). Teams will normally change their rosters slightly for playoff series, perhaps bringing in an extra hitter for a pitcher, or a pitcher more used to pitching from the bullpen.
The World Series
The playoffs climax with the World Series (also known as the "Fall Classic"), matching up the champions of the two leagues, playing a best of seven series with AL rules in AL ballparks, and NL rules in NL ballparks.
Home field Advantage for the World Series is currently decided by whichever league wins the "All-Star Game" - a serious debating point for all Baseball fans. In the past the two leagues simply took turns - one year the AL, the next the NL.