The name Genua is based on Genoa, a city in Italy. The old Latin, Germanic and Ligurian name for Genoa is Genua. However, Genua bears a striking resemblance to the French and Caribbean-influenced American South, especially New Orleans. Local transport is by paddle steamer with onboard gambling, like the early days of the Mississippi River. The local witch, Erzulie Gogol, is a Voodoo mambo complete with personal zombie (in her case, that of the former ruler of the principality, Baron Saturday, who himself shares a name with a famous Voodoo god) and the local food is basically Cajun cuisine, including gumbo.
During the events of Witches Abroad the city was ruled by an evil fairy godmother, Lily Weatherwax, (sister of Lancre witch Esme Weatherwax) though she went by the name Lilith de Tempiscire, which is French for Weather and Wax. She was obsessed with fairy tales and thought she could force everyone to be happy if only she could get them to follow the logic of the "happy ending." Under her rule, innkeepers who were not fat and jolly or inclined to tell jokes were sent to the dungeon, and Genua's ruler, the Baron Saturday, was murdered so that his heir apparent, Ella, could be a neglected orphan, as the tales dictated. Soon, the city began to resemble a skewed version of Disneyland. The name "Genua" therefore could also be an ironic echo of "genuine", which is exactly what Genua, at this time, was not. Eventually, thanks to the efforts of the Lancre Coven, Lily was deposed and sent to a dimension of endless mirrors, and Ella was able to assume her role as Baroness of Genua without, as Lily had intended, marrying a frog.
Genua was most likely established as a far flung colony of the Morporkian Empire, thus explaining why it shares more similarities with Ankh-Morpork/Sto Plains than its immediate neighbours Überwald and Muntab. The state first gained power under General Tacticus, who, having being sent there by the Morporkian Empire to stay out of the way, proceeded to declare war on Ankh-Morpork.
According to the novel Going Postal, Genua is located approximately 2000 miles from Ankh-Morpork by road, a journey of two months by stagecoach. A faster method is to travel by flying broomstick, whereby the city can be reached in a single day, but unfortunately you'll most likely blow your own ears off or possibly just freeze to death on the way.