Florence's "second" market (after the famous Mercato Centrale), the Mercato Alimentare Sant'Ambrogio offers fresh food and other goods, from around 7:00 AM to 2:00 PM every day except Sunday. Saturday mornings especially are very busy - and some vendors only come on Fridays and Saturdays. In recent years there was some experimentation of longer afternoon hours (to give people who work a chance to make it to the market) one or two days a week but I am not sure if that is still the case.
Outside on the long west side of the market you will find mostly fresh fruit and vegetables, as well as some cheeses and cold cuts, olives, bread, eggs, soaps, honey and more. On the smaller north and south ends there are clothes, shoes, and various household items (the east side is a small parking area and is used for deliveries and trash removal). Inside are meat, cheese and fish vendors, along with some general grocery, a bar and a trattotria, "De Rocco", which is a great place for an inexpensive lunch. The Macellerie (butchers) here are usually of very good quality, among them you can find specialties like horse steaks and small birds that you may not find in other spots in Florence. The lone fish vendor has a lot of frozen products - if fish is very important on your shopping list you will find more selection at the Mercato Centrale.
Overall this is our favorite food market - the experience is authentic (as it can be in a tourist town like Florence), the service is friendly, and there are (usually) more Florentines than tourists - not to mention all the great food, and some really good prices, especially on produce.
Piazza Ghiberti, near Piazza/Chiesa Sant'Ambrogio