Usually, RP 'servers' for GTA Online refer to a number of illegal multiplayer clients that allow players of GTA 5 to connect and play together while circumventing Rockstar's official Online functionality. These are in clear violation of the Terms of Service, and result in near-instantaneous bans the moment their use is detected.
However, one does not need to risk a ban in order to get into roleplaying in GTA Online. Plenty of players are interested in this kind of play, and entire crews have arisen to facilitate it within the bounds of Rockstar's official Online multiplayer space.
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So what is roleplaying? This increasingly widespread pastime is hardly restricted to gaming, and has its roots in the pen-and-paper RPG era. Various forms of non-digital roleplay, such as LARPing and various board games exist as well, and have large and healthy communities.
The point of roleplay is that you cease to be you playing some game, but self-insert and "live" the life of your character. It's a full-blown alter ego - depending on how dedicated you are. Roleplaying has a number of rules. RPing is usually restricted to a given setting or faction. Generally speaking, you can't be a Space Marine when playing Dungeons and Dragons.
Roleplaying in GTA Online has grown over the years and several distinct factions have arisen with several dedicated crews, though you'll find RP communities for all kinds of styles. Among the most popular of these are police, bikers, military, Mafiosos and gangs, though non-police emergency services are becoming increasingly popular.
Basically, players interested in one of these roles join a crew and form their character around it. You're no longer John Smith, a family man who plays GTA Online in his off-hours after work, but you become your character, who happens to be a mid-ranking member of a crime family, and you act accordingly.
If you're a police RPer, you don't commit criminal acts, but rather act like a proper police officer. If you're a mafioso, you're expected to dress sharply and protect the interests of the family. If you're a biker, you're expected to dress the like and ride. The list goes on.
Some of the more popular RP crew like to face-off every now and then in a friendly show of rivalry, while others are more than happy doing their own thing. Roleplaying is a great way to make some friends in GTA Online and to interact with a friendly community, instead of suffering with the griefers and the randoms. Luckily, the RP scene is thriving with more and more options introduced via DLCs.
If you're interested in finding a great roleplay crew that fits your interests, there is a handy Subreddit for that purpose, as well as an entire community site dedicated to GTA Online crews. There's slim chance that you won't find a community that you'll like being part of, and roleplaying will open up all new angles in GTA Online, adding new and fresh twist to an otherwise familiar game.