Game Over is a message that is displayed on screen, announcing the end of gameplay, after the player has failed one the game's main objectives. The Game Over screen can display a number of other messages as well, such as "You Are Dead" or "Mission Failed", though "Game Over" is, far and away, the most commonly used. The idea is now so universally known that it has become a common phrase in the English language used to signify the end of something.
Like many of the classic video game mainstays, the concept of the Game Over was constructed out of the necessity for arcade games to get more money from players. The idea was to limit gameplay time so that people could not simply continue playing forever on a single quarter, thus making more money for those collecting.
The game over screen, classically, was shown to tell the player that they had failed the game (or in some cases, finished the game), and could progress no further. Their only option being to insert more money into the machine and begin playing the game again from the very start. This later gave birth to the "Continue?" screen which gives the player the option to continue progressing through the game after losing, provided they insert more money into the machine.
Games were usually constructed in a manner in which the player would have a limited number of tries to complete a goal or beat the game before their play was ended by the getting the Game Over screen. The most common of which was the concept of the player having a finite amount of "extra lives". One extra life would be taken from the player each time their character died in the game, until all extra lives were gone, resulting in Game Over.
In the era of the home console, the Game Over screen is an increasingly rare sight in modern video games. Because most games are now a one-time purchase, there is no need to force a final end to the gameplay onto the player in order to get more money. This allows games to be longer, have more depth and complexity, and better stories, among many other things.
Many games still feature the Game Over screen for when a player has failed one of the games main objectives, but rather than forcing the player to restart the game from the beginning, the Game Over screen now more commonly presents the player with a series of options which could include restarting from a checkpoint, from the beginning of the level that they're currently playing, from the most recent point at which the player saved their progress, or simply quitting out of the game.
However, many modern games have completely done away with the concept of the Game Over screen, instead continuously reverting to the most recent checkpoint or save point each time the player fails until they choose to quit.
Further, because video game designers no longer rely on limiting gameplay to gain profit, the door is opened for many genres of games to be made that could never exist in an arcade setting because they do not feature any way to fail or any real end to gameplay, allowing a person to continue playing a game for as long as they like.