For a quick bite, go to a bar -- though it does serve alcohol, a bar in Italy functions mainly as a cafe. Prices at bars have a split personality: al banco is standing at the bar, while à tavola means sitting at a table where they'll wait on you and charge two to four times as much for the same cappuccino. In bars you can find panini sandwiches on various rolls and tramezzini, giant triangles of white-bread sandwiches with the crusts cut off. These run 1.50€ to 5€ ($1.95-$6.50) and are traditionally stuck in a kind of tiny pants press to flatten and toast them so the crust is crispy and the filling hot and gooey; microwaves have unfortunately invaded and are everywhere turning panini into something that resembles a very hot, soggy tissue.
Pizza à taglio or pizza rustica indicates a place where you can order pizza by the slice, though Florence is infamous for serving some of Italy's worst pizza this way. Florentines fare somewhat better at pizzerie, casual sit-down restaurants that cook large, round pizzas with very thin crusts in wood-burning ovens. A tavola calda (literally "hot table") serves ready-made hot foods you can take away or eat at one of the few small tables often available. The food is usually very good, and you can get away with a full meal at a tavola calda for well under 15€ ($20). A rosticceria is the same type of place with some chickens roasting on a spit in the window.
Full-fledged restaurants go by the name osteria, trattoria, or ristorante. Once upon a time, these terms meant something -- osterie were basic places where you could get a plate of spaghetti and a glass of wine; trattorie were casual places serving simple full meals of filling peasant fare; and ristoranti were fancier places, with waiters in bow ties, printed menus, wine lists, and hefty prices. Nowadays, though, fancy restaurants often go by the name of trattoria to cash in on the associated charm factor, trendy spots use osteria to show they're hip, and simple inexpensive places sometimes tack on ristorante to ennoble their establishment.
The pane e coperto (bread and cover) is a cover charge of anywhere from .50€ to 10€ (65¢-$13) that you must pay at every Italian restaurant for the mere privilege of sitting at the table. Most Italians eat a full meal -- appetizer and first (primo) and second (secondo) courses -- at lunch and dinner and will expect you to do the same, or at least a first and second course. To request the bill, ask "Il conto, per favore" (eel con-toh payr fah-vohr-ay). A tip of 15% is usually included in the bill these days, but if you're unsure, ask "è incluso il servizio?" (ay een-cloo-soh eel sair-vee-tsoh?).
