Shared-author fiction is where a story is told (or written) in successive pieces by different authors. It is distinct from other forms of collaborative writing in that each piece of text is written by a single author. Thus, a simplistic shared-author piece might be made by two people where one person writes odd-numbered pages while the other writes even-numbered pages. The story is written in order, where each author must pick up where the last one left off and continue in a coherent narrative.
This becomes role-playing when each author has associated characters which they own. Then each author may write a section about what their character does. These are role-playing games of a sort which these days are typically played online using web-based bulletin boards or other internet forums to post the story text. Unlike most computer games or tabletop games, a true shared-author game has rules structured as etiquette and style guides rather than means of resolution (i.e. "no curse words", "no sexual situations", etc.). Often the author of a given character has final word over anything that seriously impacts that character. i.e. A character can only be wounded or killed with that player's permission.
Shared-author/role-playing sites frequently are based on science fiction and fantasy television series -- evolving out of the phenomenon of "fan fiction". There are numerous Star Trek shared-author RPGs, for example, along with Xena, Buffy, and various Japanese anime series.
Really, there is no clear line between shared-author RPGs and others. The form is distinct mainly in that it creates a (hopefully) coherent text narrative as its product. However, it certainly can overlap with tabletop play and online computer games. Some tabletop games are played over email (called Play-By-Email or PBEM). I list below a few links to what I would consider shared-author roleplaying games: