Palazzo San Giorgio was built in 1260, by order of the Capitano del Popolo Guglielmo Boccanegra, for the Commune. In 1570 it was extended and this frescoed facade, showing Saint George and his dragon, was added. Later, in the 15th century, the palace became the headquarters of the first European bank, Banco di San Giorgio.
Marco Polo was actually imprisoned inside when it was the Customs House, and it was there, or so it is told, that he dictated his account of his travels to China to a fellow prisoner.
Source: the truly excellent guide book, in English, titled Genoa Guide, by Sagep Turismo. If visiting Genova, I would recommend you buy yourself a copy, as I love it and find myself devouring it like a good novel.