PBEM stand for "Play By E-Mail". It's when two or more people communicate what they're doing with each turn of a game via E-Mail to the other player(s). It takes longer to play this way than FTF gaming, but as gamers get out of college and into real jobs, real families, and real lives, it is often the only way they can get to play any of their old favorites other than once or twice per year at gaming conventions.
The enormous amount of information that has to be tracked, and the amount of time it takes to translate movements on a game board to E-Mail text, have led to the development of a number of support tools, most notably those which permit a graphic recreation of critical elements of the game (map and pieces, at the very least). Many of these tools actually come with graphical interfaces where the players actually move pieces around the board and a small encoded description of the moves is sent to the other player(s), who load these "moves files" into their copy of the program and they watch their opponent's moves play out on their own computer.
Some gaming companies have been gracious about letting fans of the games create and distribute electronic versions of key game components, while others have not. But for most hardcore gamers, PBEM is the only mechanism by which they can still enjoy their hobby.