It was founded by Albus Dumbledore in 1970 when Voldemort started his first attempt to take over the Ministry of Magic-- beginning the Dark Lord's Anti-Muggle campaign. The Order of The Phoenix lost many of their members by the time Voldemort was defeated by Harry Potter, in 1981. It was later refounded after Harry informed Dumbledore of Voldemort's return in 1995. The Order does not have as many members as the Death Eaters, and as such many of their members have significantly more screen time and characterization.
They appear in many stories due to the large roles they play from the fifth book on. It can be considered a charge to add or detract members from the Order, both past and present, without a very good reason (such as well-done AU). The Order had to be operated in secret during Voldemort's second rise to power due to the Ministry not admitting his return; forgetting this is a definite charge.
Due to their smaller head count than the Death Eaters, they are rarely given the same roles as canon fodder. There are rarely 'generic Order members.' On the other hand, this can lead to them being either completely ignored, or them only being mentioned as background characters. They are often displayed as inept when left to guard Harry, or, in rarer cases, evil, following the orders of Evil!Manipulative!Dumbledore. They are also often shown as being weaker than in canon, and need to be saved by the Sue or Replacement. Due to their large trust in Dumbledore, they may also be unwilling pawns used by him, whether he is evil, or just misguided.