Actually, I misspoke (or mistyped). What we would call a cafe is called a "bar" in Italy. Don't worry - while they do serve alcohol, they also serve a variety of non-alcoholic drinks. No one is there to get hammered. This is where Italians get breakfast (a coffee and cornetto, similar to a French croissant), and where they stop in all day (into the early evening) for coffee, pastries, sandwiches, and light snacks. You can stand at the counter to eat and drink. If you want to sit down, you MUST do this first, and you will pay more; the table is then yours for as long as you like. It is a big no-no to order at the counter and then take your order to a table (like many others, I did it on my first trip because I didn't know better).
If you're getting counter service and not sitting down, it's also common at bars (and gelato places) for you to have to pay first at the cashier and get a ticket, then place your order and hand in your ticket. This way, the servers don't handle money. It's common for us non-regulars (who don't know by heart what's on offer) to have to go to the food area to choose what we want, then go to the cashier to pay for it, then return to the food area to actually get it. This is not hard once you adjust, but it is different from the usual US system. Just watch others and imitate.
A restaurant is a place with table service only. In Italy, a restaurant meal is an event and takes a while (90-180 minutes). Again, they do not expect to "turn tables"; once you are seated, the table is yours for the evening. So, if there's a particular restaurant you want to eat at, make a reservation (even if it's just dropping in at lunchtime to make a reservation for dinner). If you don't have a reservation, get there when the restaurant opens, and you can hope to snag one of the few unreserved tables. Restaurants are often closed for several hours between lunch and dinner. And because they only have one seating per table per meal, you won't get served if you get there too late (in other words, if they're open from 1-3 for lunch, you won't be served if you get there at 2:30 - I know because I've tried!).
Once you're in Italy, the difference between a bar and a restaurant will be very obvious. What I meant was, a bar, since it specializes in drinks, will have the fresh fruit and machine (or squeezer) to make a spremuta, while a restaurant usually will not.