This page deals with the multiplayer concept of resetting one's progress after reaching the maximum experience level, typically earning a cosmetic bonus or icon that lets other players know that the player in question is on his or her second (or third, fourth, and so on) trip through the leveling system. Prestige mode gets its name from the idea that players can choose to trade in their progression for a badge of supposed "prestige" in the eyes of other players.
The concept was popularized by Infinity Ward's Call of Duty 4: Modern Warfare. As games leaned into COD4's persistent progression as the template for their own competitive multiplayer modes, the concept of "prestiging" made its way into many other games, often using the exact same term.
For a single-player take on this concept, see New Game Plus. New game plus generally refers to resetting the state of the game world and story, and it may occur with or without resetting player character progression.