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Here are the typical coffee drinks you will find in Italy:
caffè (espresso): a small cup of very strong coffee, i.e., the typical espresso (20 to 25ml)
caffè doppio: 2 espresso served in one cup
caffè ristretto: an even more condensed version of an espresso (less than 20 ml)
caffè macchiato (or un macchiato): an espresso with a drop of milk. You can order either a macchiato caldo (drop of steamed milk), a macchiato freddo (drop of cold milk) or a macchiato con schiuma di latte (with froth milk).
caffè Americano: weaker than espresso and served in a large cup or a glass, but still stronger than American-style coffee. It consists of an espresso to which hot water has been added after the brewing process (about 80ml). It is different from a caffé lungo (see below).
caffè lungo: where more water (about double) is let through the ground coffee, yielding a weaker taste (about 40ml).
cappuccino: it is not just an espresso with steamed milk, although it may seem that way to the casual observer, but if you look more closely next time you have one prepared in an Italian bar you will see that they pour the steamed milk very carefully as to add about 1/3 of steamed milk before the final topping (of about 1/3) of foamy, frothy milk. Served with or without the finishing touch of cacao powder or chocolate topping. Italians usually drink cappuccino in the morning only, never after lunch or dinner. The name Cappuccino comes from the resemblance of its color to the robes of the monks of the Capuchin order. Note that a cappuccino is often called a cappuccio, especially in Southern Italy.
Some like the idea of a cappuccino, but prefer it with a stronger coffee taste. In that case there are two options: either you order a cappuccino con doppio caffé, which is basically a caffé doppio prepared as a cappuccino, or you try a cappuccino scuro: which is intermediate between a cappuccino and a caffé macchiato, i.e. with the same amount of coffee but proportionally less milk than a standard cappuccino.
caffè Hag: also more and more called un deca, a decaffeinated coffee. Can be combined with any of the versions above, un deca macchiato, un cappuccino deca, etc.
caffè con schiuma (or caffè schiumato): an espresso prepared in such a way to produce "schiuma" on top of it.
caffè corretto: an espresso with a drop of liquor.
caffè borghetti: an espresso with a drop of borghetti liquor, which is typically consumed at break-time in the soccer stadium.
caffè freddo: espresso shaken with ice and sugar and served in a glass, usually in summer.
caffè con panna: espresso with cream on top, what the rest of Europe usually calls a cappuccino.
caffè latte: not really a coffee drink, as this is basically hot milk mixed with coffee and served in a glass.
gran caffé speciale: an extremely creamy double espresso, served only at café Sant Eustachio in Rome, a place renowned for serving the best espresso in town.