The button is essentially a person's "off switch" in boxing. It's located as close to the point of someone's chin as you can get and still land a solid punch (see diagram).
Why as far out as you can get it? Remember back to your simple machines. The further away the punch lands from the neck, the more torque there will be and it will be harder for the punchee to brace themselves against the punch. You'll whip their head around faster/further and their brain does the little "wup wup wup wup" inside their skill and they get a concussion, leading to the knockout. (See What's in a Knock Out? for more details on what actually causes the KO)
Most commonly, hooks and overhands (especially in MMA) are used to hit the button.