Mr. Game & Watch is the playable character of several games and mascot of the Game & Watch series. He is not formally named until Super Smash Bros. Melee and it is not known whether each game is the same or different person, but base off his moves in the Super Smash Bros. series it is assumed that they are all the same person.
In each game Mr. Game & Watch has a specific job to complete. In his first game, Ball he is a juggler and must not drop the balls he is juggling. Some other jobs include a flagman, fireman, city worker, lion tamer, sky diver, chef and a greenhouse worker. Mr. Game & Watch has made cameo appearances in other games such as the shop keeper in Wario Land 4 and working in the background in Donkey Kong Country Returns. He makes a larger appearance in the Wario Ware series where several of his games have been converted to the mini-games featured in the series.
"The monochrome hero of the Game & Watch series, a fellow who oozes personality. First appearing in 1980, the series were the original portable game systems. The first games in the series were quite simple, but as time passed, technology evolved. The multiscreen series featured game play on dual screens. As the name states, the games also kept time." - Mr. Game & Watch Trophy, Super Smash Bros. Brawl
TL;DR He's a generic faceless character whose voters consist entirely of Smash Bros. fans.