As per Wikipedia:

The game itself, originally called Bingo, is thought to have had its roots in Italy in the 16th century, specifically, around 1530.Bingo originates from the Italian lottery, Il Gioco del Lotto d'Italia. From Italy, the game spread to France and was known as Le Lotto, played by the French aristocracy. The game is then believed to have migrated to Great Britain, and other parts of Europe in the 18th century. Players mark off numbers on a ticket as they are randomly called out, in order to achieve a winning combination. The similar Tombola was used in nineteenth-century Germany as an educational tool to teach children multiplication tables, spelling, and even history.

The origins of the modern version of the game, and its current name of Bingo, are unclear. Early British slang records Bingo as... "A customs officers' term, the triumphal cry being employed on a successful search". But it definitely gained its initial surge of popularity with the first modern version of the game appearing at carnivals and fairs in the 1920s, and is attributed to a Hugh J. Ward, who for marketing reasons most probably took the name from pre-existing slang. The patent for a modern Bingo card design went to Erwin S. Lowe in 1942.

A more comprehensive read on Wikipedia.