A 'subway series' is a series of consecutive baseball games played between teams which are both based in New York City, and hence can all be attended using no means of travel other than the New York subway system. The term originally was used to refer only to those World Series championships which took place between New York teams; however, with the advent of inter-league play during the regular baseball season in 1997, sets of such games between New York teams have also come to be referred to using the term.
Although a 'World Championship Series' took place between New York and Brooklyn in 1889, this doesn't count - for one thing, the term wasn't used, for another Brooklyn wasn't yet part of New York City, and for a third the subway didn't exist yet. The term first cropped up in newspapers in 1927, although the first such set of games had taken place between the New York Giants and New York Yankees back in 1921 (when both teams were New York teams, the subway existed, etcetera).
The New York Yankees, being the only American League team in New York, have been a contestant in every Subway Series to date. The real rivalry was the Brooklyn Dodgers versus the Yankees - and the Yanks won every subway series until 1955, when Brooklyn overturned the Bronx Bombers in a seventh game shutout. Nowadays, with the Dodgers and Giants gone, the New York Mets are the only team left to hold up the torch, and they haven't fared well against the massive payroll in The House that Ruth Built, either.