Either a text-based or graphical game in which the player-character has a name, a background, and a history in the game world, and for whom the game has some particular physical skills and training in the use of objects (overwhelmingly weapons) programmed in for the player to command.
More recent text-based adventure games have often been first person, especially those in which part of the game's task is determining the player-character's place in the game world. These games require the player to act (type in commands and statements, really) that would be appropriate for the player-character to do in non-interactive fiction. The game can be programmed to punish the player-character if the player types in some inappropriate responses.
The graphical games of this sort present the player-character as merely a pair of arms holding and aiming objects on the lower part of the screen. Few graphical adventure games are first-person games. If the story requires a player-character distinct from the player, then a third-person presentation is used to emphasize the distinction.
All first-person shooters (FPS) are this sort of game, by definition.