I second Vini e Vecchi Sapori which truly is fab and very simple; as is the Buca del Orafo - food is truly very simple, good quality ingredients which are well cooked. You will hear many Italian voices at both only punctuated by the odd English American. I also rate the Osteria de Benci and the Osteria San Niccolo which I would rate over Trattoria 4 Leone anytime (go to San Niccolo if you would like to spend the evening in the Oltarno). Il Latini's atmosphere is amazing, but food is over rated. Casalinga near Santa Spirito is also good, total hole in the wall but very reasonable. Il Santa Bevitore more upscale is also very good. If you are heading to San Miniato or Piazzale Michelangelo go to Fuori Porta
I Latini is not the greatest cuisine, but many visitors like to try it because you sit on benches at large tables, there is a lot of noise and action, the food is OK and the price is (at least was) reasonable. Many shy away from it because it tends too attract too many tourists. On our only visit there about 8 years ago, there seemed to be more English spoken than Italian by diners.
Acqua al Due is appealing, but may be hard to get into and alos suffers from an abundance of American tourists.
Sostanza is worth a visit for the food (not much atmosphere, but you go there for the food). Try several of their regular dishes: bistecca fiorentina, petto de pollo en burro (don't miss it!), and their artichoke dish.
Not sure what you are seeking: is price an issue, is authenticity desired? Vini e Vecchi Sapori is a small and inexpensive place but serves very good homemade dishes (small restaurant, not a lot of atmosphere, not chi-chi, but very authentic). We always like the gnocchi in black truffle sauce and the pici senese pasta at Antico Fattore.