Not only does this restaurant have a wonderful atmosphere (they do a an 'artisans lunch' which brings the locals in), but the food is absolutely delicious.
They are most famous for their Fiochetti alle Pere con Salsa di Taleggio e Asparagi (here's the recipe on their website! http://www.4leoni.com/ricette/fiocchetti-alle-pere-con-salsa-di-taleggio-e-asparagi/2.htm) and Peposo all'Imprunetina, which are both eye-wateringly delicious, and which you can eat sitting under an umbrella in the piazza.
This is the perfect place to eat out when your family come to stay or when you want to treat yourself. It's not very expensive, but it's not student food. It is, in fact, my number one restaurant recommendation in Florence.