Democracy is theoretically a game mode that helps players overcome the trolling malignancy of the stream by allowing a more judicious way of governing the character. Chat comments are translated into actual button inputs and directional commands like usual, but through a voting system that selects an input every 20 seconds and then resets. While terribly slow and overall more ineffective, it's a far more careful way of performing complex maneuvers, which in Twitch Plays Pokemon can be something as simple as opening a menu and selecting the necessary option while facing the right way.
On the other side is anarchy, a game mode that retains the original channel's mechanic of an all-out free-for-all where any one of the thousands of game commands flooding in every minute can form the basis of in-game movement.
In "Democracy," the majority rules -- however most people vote rules the character's actions.
In "Anarchy" -- real anarchy -- the character would make his own decisions, proceeding unruled by the authority of others.
The "Anarchy" mode in "Twitch Plays Pokemon" is misnamed -- it's actually more like what America has been becoming over the last few decades: The character finds himself subject to the arbitrary, confusing and conflicting commands of every politician, bureaucrat or badge-holder who might, at any moment, take an interest in ordering him around.
The only difference is that the character doesn't have the option of defying these commands (or, fortunately, of being beaten to death for QUESTIONING AUTHORITAH as per Semi-Standard Operating Procedure for "law enforcement").