The "Ban Button" is a
hypothetical button (though some network administrators and
moderators actually have a key assigned to said feature), that more than often refers to banning someone from something for some particular reason.
The "Ban Button" can refer to a number of things, whether it be a button on a forum or game that
bans, or a key that bans.
The origins are not entirely sure of.
Person 1:
I'm trolling you!
Admin: *Presses the "Ban Button".*
Person 1: Aww! I'm so angry now I'm going to threaten to
take down this site with my bare hands yet I will not do anything because deep down inside
I'm scared!