Only an hour from Florence by train, Pisa should be high on your list of sights to see in Italy. The most iconic structure to see is Torre Pendente or the Leaning Tower, which was conceived as the bell tower for the splendid Duomo di Pisa (or Pisa Cathedral). Shortly after construction began in 1173, the tower started to lean due to the soft ground on which it was being built. Tower construction occurred in 3 phases over a period of 199 years.
Galileo Galilei is said to have dropped two cannonballs of different masses from the tower to demonstrate that their speed of descent was independent of their mass. It’s not known if this is completely true or not – but it does make a good story and demonstrates an important property of physics
During World War II, the Allies suspected that the Germans were using the tower as an observation post. A U.S. Army sergeant sent to confirm the presence of German troops in the tower was impressed by the beauty of the cathedral and its campanile, and thus refrained from ordering an artillery strike, sparing it from destruction. (you can read an account of this story here)
Numerous efforts were made to stabilize the tower and it was closed for a long time for safety reasons. In 2008 it was reported by engineers that it stopped moving for the first time in 800 years and that it should remain stable for at least another 200 years.