A continue is the common term in video games for the option to continue the game after all of the player's "lives" have been lost, rather than ending the game and restarting from the very beginning.
In home console games, continues are not typically available at the player's leisure; he must earn them, either by collecting an item, reaching specific point totals, or performing certain tasks. Furthermore, continues may not place the player directly at the point they ended. For example, in a platform game, an extra life may allow a player to continue from the point their character died, but a continue would leave the player at the beginning of the level he was playing.
Most modern arcade games always contain a "continue" feature whenever a player loses all of his or her lives, but must use another credit (insert more coins) before it is allowed. Typically, the player is given a short amount of time (usually ten seconds) to continue before "game over."