The world border is introduced in Minecraft 1.8, replacing the static and solid edge of the world (aka the Far Lands) from 1.7.2 with a configurable border.
Some time after the server's update to 1.8, Molster set the border of the server to correspond to his map renders. He argues that it is to discourage aimless exploration and encourage players to actually build, adding that building so far away is pointless as people aren't going to be able to look at them without having to rely on transportation infrastructures which took time to develop.
This border was extended to 1 million blocks across on late-July 2015, effectively removing it. This is to give more freedom to players and restore access to cut-off areas, though any future map renders would still be constrained by the old dimensions.
The border is centered at (255,255), the center for region (0,0).