Anarchy is pretty much what we don’t really want, we don’t want lawlessness and no rules which is why most games have a pretty rigid system of rules governing how and why to play them, what makes them work and most importantly how to win at them. Minecraft being the trailblazer that it is among games, had none of those. There are basically no rules to speak of off, on the earliest servers you could play in any way your heart desired, whether it was building and creating the largest buildings and masterpieces, whether it was roaming the land and underground in search of rare diamonds and finding what you could find in the vast world around you, or whether it was fighting other players and mobs found in your server, trying to show the world that you were among the best at fighting. Minecraft anarchy servers basically take that idea and runs with it to the extreme. They open their servers to anyone willing to play on them without any rules or obligations, save refraining from cheating and hacking the servers systems, they keep the lawless essence of Minecraft and try to expose the players to it as much as possible. On anarchy servers people aren’t going to coddle you and you wont be able to call the mods/admins when your base gets wrecked by a person you pissed off by mining their diamonds in front of their face, while they were getting chased off by a creeper.
There are plenty of ways to play Minecraft, however anarchy truly stays true to the heart and soul of Minecraft being that it’s an actual sandbox game, letting you do whatever the game engine let’s you do. It might not be as popular as other game modes because of it’s rather eccentric and sometimes downright hostile communities, however if you can bridge the storm at the beginning you’re going to find a really worthwhile experience at the other side, with basically what no other game mode offers, a true Minecraft experience.