I Would Like To Join Your Game!
Lady S mentions system and freeform games, so I thought I'd drop a few words about those terms.
Freeform games are exercises in creative writing where the GM (Gamemaster/mistress) lays out the situation/world/plot in greater or lesser detail and the players write about their characters' actions and thoughts and feelings. Conflict is described narratively with the GM making having final discretion (usually) on the outcome. Rules are few or none, mechanics are limited, and die rolls rarely employed.
System games have more structure and are generally meant to use a particular published rule set and often a pre-established "world/universe" in which the story plays out. Some rules systems are more complicated than others, but most involve using die rolls to determine character success or failure. Dungeons and Dragons is probably the best known of these systems in several different and widely varying editions, some running to dozens of supplementary books, but there are many, many other systems.