Rob,
Baseball is divided into two diferent leagues: the National League and the American League. The champions of each league meet in the so-called World Series, which has been played since 1903. The Mets play in the National League and the Yankees in the American League. Both are from New York.
The Yankees are by far the most successful team in the history of the sport, with 26 World Series Championships to their name. The next closest team are the St. Louis Cardinals with 10 and the Boston Red Sox, with 7. The Mets, but 2
The Red Sox are the archrivals of the Yankees. Until 2004, the Red Sox had not won a World Series since 1918. Legend has it that when the then owner of the Red Sox sold Babe Ruth (who would change how baseball was played) to the Yankees so he could raise money for a Broadway play, a curse was put on the Red Sox, the so called curse of the Bambino (referring to Babe Ruth's nickname). The Yankees and the Red Sox, no matter how good or bad they are, are antagonists, with the Red Sox usually suffering the worst from the Yankees.
For example in 1949, the Yankees went to Boston to play a 3 game series at the end of the year. The winner would go to the World Series. The Red Sox had a two game lead and only needed to win one game; they couldn't win even one. In 1978, the Red Sox in July had a 14 game lead on the Yankees. Normally you would say the race was over. However, the Yankees caught up while the Red Sox kept losing and losing and in a one game playoff (they finished the regular season tied), the Yankees beat the Red Sox.
However, in 2004, when the Yankees were playing the Red Sox for the championship of the American League, the Yankees were up 3-0 in games and only needed one more to win and since, after all, no team had ever come back to win 4 in a row, it was pretty much of a foregone conclusion that the Yankees would win the title and go to the World Series. The improbable happened: the Red Sox proceeded to do that and went on to win the World Series. The Babe Ruth curse had been broken.
Hope that give you a little flavor for the great game this is.