A forum role-playing game is a game where players assume roles of themselves-created characters in a given world (most often fictional), and write posts - passages of story - on their behalf to further the game. An FRPG usually contains a detailed description of the given world and one or more main plotlines.
Each player may wander aimlessly on the given world territories (usually separated in categories, with each place preserved in a single forum topic) doing stuff they deem fit for them or take part of personal or plot-related quests; neither type of quest is limited to a single character unless that's the point of the quest (which is rare anyway: the main reason people take up FRPGs is literary cooperation with other players' characters). Rewards for well-played quests are usually in-game items or medals which distinguish the characters or the players but don't give them significant advantage over other players' characters.
Players are usually balanced by some strong barriers (like: your character is too young to take up so many advanced skills) as well as some weak ones (like: your character doesn't fit the theme of the game), with weak barriers defined on case-to-case basis by the administration (which usually consists of the authors of the world and who may or may not take part in the game as well).
Think of it as writing a book, with each more-or-less-major event being written by a different author, from different perspective. Would you take part in such an activity?