New Game+ (pronounced "plus", and known by a variety of other names, such as New Game EX, New Game X, etc) is a mode, traditionally unlocked upon completion of the game, which allows the player to start a new game while carrying over things (e.g. level, items, money) from the previous playthrough. This concept is typically seen in role playing games, but other genres have used the concept as well. New Game+ is usually utilized in games which contain multiple endings, so that players can attempt for another ending.
The option itself dates back decades, but the term "New Game+" wasn't actually used until 1995's Super Nintendo RPG, Chrono Trigger. In Chrono Trigger, a game with multiple endings, using New Game+ was a requirement to see the vast majority of these endings.
Ordinarily New Game+ allows for the carrying over of items, money, and experience, although there have been a wide range of variations on the concept. Special items or modes can sometimes be found in New Game+, such as Sigma Star Saga, the various Mega Man Battle Network games, and X-Men Legends 2: Rise of the Apocalypse.
Other games, such as Castlevania: Dawn of Sorrow, allow players to control a different character during New Game+. During this mode, experience, money, and items are typically not carried over. Still other games allow players to unlock new weapons or upgrades, such as Resident Evil 4 and Resident Evil 5. The Ratchet and Clank franchise has the unique option of playing through the game in first person mode.
Another variation of New Game+ that deviates from the more common definition is called Clear Mode (or "Clear Game"). In this mode, the game continues past the true final boss of the game, allowing players to essentially see the effects of finishing the game. Some early games that offered this option are Dragon Warrior III and EarthBound.