The King's Justice is the title held by the royal executioner. His duty is to kill anyone condemned by the King, the Hand of the King, or the small council, usually by public beheading.
The current King's Justice is Ser Ilyn Payne, a knight formerly in service of Tywin Lannister and whose tongue was removed under orders of the Mad King.
Other uses of the term
The term "King's justice" is also employed to refer to the manner in which lords carry out justice in the name of the King. The King or the Hand may send lords or knights to "bring the King's justice" (capture or death) to brigands or anyone who disrupts "the King's peace."
In the Eyrie, the term refers to the Moon Door, a hatch-like door in stronghold's great hall, which opens to the hollow bottom of the mountain in which the castle rests. The Moon Door is used to carry out executions instead of employing headsmen.
Meanwhile in the North, where the customs of the First Men are still followed, the lords themselves carry out the King's justice, due to the belief that whoever passes a death sentence should also swing the sword.