Restaurants in Livorno, as in the rest of Italy, come under various headings, and are not all just plain old "Ristoranti". You'll find "Osterie" and Trattorie" too, as well as "Pizzerie" and "Enoteche", and Livorno has a good choice of all of these.
Traditionally, when eating out in Italy, the osteria and trattoria are cheaper and more popular alternatives to the smarter ristorante. However, this is not always the case any more and it is probably more helpful to have a look at a menu and peer through the window to see what a place looks like! White linen tablecloths often indicate a smarter, more formal, and therefore more expensive restaurant, but not always, as this kind of table linen can also be indicative of simple traditional cooking.
The term Osteria is quite fashionable these days, so it may not always indicate a cheaper place to eat. A trattoria, generally speaking, offers traditional informal dining. In Livorno for example, the Trattoria Da 11 is a popular informal restaurant in the San Marco district.
Pizzerie of course speak for themselves, but they often offer meals other than pizza too. In Livorno you will often see signs saying 'Pizzeria Ristorante'.
Examples of this in Livorno are:
Chez Ugo, Pizzeria & Restaurant, by the canal in the old Venezia district.
An Enoteca or wine bar will usually still serve meals, but there will be a very good wine cellar too.
The Cantina Nardi is a good example.
Italians don't normally go out to drink wine without something to eat, and even the sparkling prosecco often served as an aperitif will be served with some kind of savoury snack, even if it's just a few peanuts!
In Livorno, the aperitivo is very popular, especially in the evening. Far from a dish of nuts, the bars and cafés who know how to do this well serve an amazing selection of hot and cold snacks which can easily replace dinner. The price is around 5-6 euros including your drink! One of the best aperitivi in Livorno is served at the central Carolina Café