Play by mail games are games, of any type, played by either postal mail or email. Chess is an example of a game which has been played by mail for centuries. The wargame Diplomacy was played by mail in the 1960s, starting with an email newsletter (called a \'zine) written by John Boardman. More complex wargames, moderated by computer programs, were pioneered by Rick Loomis and his company, Flying Buffalo Games (http://www.flyingbuffalo.com/).
Today, many games are played by email as well as postal mail. In addition, some computer games can be played in a play by mail mode: you make your "move", mail a file to your opponent, who makes his "move", and mails something back.
So what's the difference between these games and the average multiplayer shoot em up game? Well, with most computerized multiplayer games, the players have to be online at the same time. With a play by mail game, the players can play whenever they choose.
For further information:
http://www.pbm.com/~lindahl/pbm.html (Play by mail homepage)