In The Godfather, Michael Corleone murders two enemies, then flees from New York City to hide in Italy. He goes to the a small town in Sicily and meets a pretty woman named Apollonia. He approaches her father at a bar and asks for permission to court her. You can watch the scene here.
The scenes were shot in the Sicilian town of Savoca. The bar is a real bar called Bar Vitelli. It still exists and remains open. You can see photos of it, inside and out, at Messy Nessy Chic.
This video shows Bar Vitelli and the surrounding streets in Savoca.
It's been many years since I've seen The Godfather.