The Neapolitans say they invented it, and it's best not to disagree, though there may be a Greek or Middle Eastern influence behind it (pita, etc). Neapolitan pizzerie certainly make some of the best anywhere, nearly all with a thin, crisp base. American bready, deep-pan pizzas, though, have their origins in the Old World too, as with the Sicilian sfincione.
Pizza al taglio refers to the ubiquitous shops that sell pizza 'by the slice'. Don't neglect them, the best ones can be amazing. They are a real speciality in Lazio, and are often baked in large rectangular pans, with a slightly thicker crust than the Neapolitan version.
There are endless varieties of pizza besides the classic marinara, including:
pizza bianca: Generally this is just with just olive oil and salt, and sometimes rosemary.
pizza Bonci: made in the style of Gabriele Bonci of the Pizzarium in Rome, with a thick sourdough crust.
pizza alla casalinga: square pizza, topped with mozzarella, crushed canned tomatoes, chopped garlic and olive oil.
calzone: pizza folded over before baking.
campagnola: 'country pizza'- everyone has a different interpretation. Sometimes with greens and mozzarella, sometimes with pecorino and eggs baked in the dough, sometimes with sausage. Best to ask before ordering.
capricciosa: Sicilian, with tomato, mozzarella olives, prosciutto, mushrooms, aubergine, artichoke hearts, and maybe a bit of würstel, according to the caprice of the pizzaiolo.
pizza chiena: ('stuffed') this one is really a rich bread, with chunks of salame and cheeses mixed in. It's made in Campania and elsewhere in the south as an Easter treat.
pizza del contadino: a kind of calzone, stuffed with escarole, pecorino, mozzarella and anchovies
alla diavola: the closest you'll get to an American pepperoni pizza, with spicy Calabrian salame.
piazza farina gialla: made with corn flour
pizza con foglie, di grandini, or di Randini: made with corn meal: see mbaniccia.
pizza frutti di mare: topped with tuna, shrimps and mussels.
genovese: a thicker, focaccia-like pizza with various toppings.
mare i monti: usually with shellfish and mushrooms
montanara: small fried pizza with a little tomato and basil on top—southern street food
panzerotto: small filled pizza
parigina: a thin calzone with one or both crusts made of puff pastry
pugliese (or barese): a focaccia with potato and cherry tomatoes.
quattro formaggi: with mozzarella and three other cheeses.
pizza rosso: with tomato sauce.
pizza rustica: more of a sandwich, or sometimes baked entire as a pie.
pizza sbattta: a cake, similar to pizza di pasqua alla romana
siciliana: Sicily-style, with aubergine/eggplant
viennese: mozzarella, tomato, and German sausage.
pizza in brodo in the Abruzzo is not pizza soup, but a broth with chicken giblets, eggs and parmigiano.
pizza di crema in Naples is a dessert made with pasta frolla and filled with custard, orange zest, cinnamon and cherries.
As much as the Italians love their pizza, when something's a bore, it's 'una pizza'.
Images by: Ron Dollete, cucina italiana