Per Major League Baseball's collective bargaining agreement, players must meet certain criteria in order to be eligible for postseason play.
Players must be on the 40-man roster, the 60-day disabled list or the bereavement/family medical emergency list as of midnight ET on Aug. 31 to be eligible for their respective clubs' postseason rosters. Consequently, players that are acquired via September trades or signed as free agents in September are ineligible for postseason play.
Clubs can petition the Commissioner's Office to add any player that was not on the 40-man roster as of Aug. 31 -- provided the player was in the organization on Aug. 31 and is replacing a player who has spent at least 60 days on the 60-day disabled list. The Royals were able to use this tactic to add Brandon Finnegan to their roster during the 2014 postseason, despite the fact that Finnegan was not on the 40-man roster as of Aug. 31.
Additionally, all players who have served part of a suspension for performance-enhancing drugs in a given season are ineligible for postseason play that year.