The history of PBM games goes back at least to correspondence chess, mentioned in some ancient and medieval records, and well documented from the 1800s when specific correspondence rules were codified. In the heyday of Avalon Hill board wargames there were similar accepted standards for playing against remote opponents by mail. That's one reason that wargame maps have had a unique ID number in every hex since at least the 1960s-70s.
In the 1970s, Flying Buffalo and Schubel & Son introduced a number of games designed specifically for PBM, primarily with fantasy and sci-fi themes, usually involving building empires and conquering the neighbors, but also including some smaller scale adventures like dungeon crawls. These games developed their own subculture, spawned many imitators, and were loved by enough players that some are still being played today in some form.
Originally players filled out a short form on paper or a postcard, or even a computer punch card, and mailed it at regular intervals to the host company. Games could be human- or computer-moderated, but they could only run smoothly if their widespread player bases replied promptly and clearly, as noted in Ben Sargent's "Murphy's Rules" feature in that same Space Gamer issue:
As a variant of PBM, Play by Email (PBEM) games appeared as soon as email access reached a sufficient population. There were some early adopters in the 1980s but it really took off in the 90s, contemporaneous with the growth of early online MUDS.