Not long ago, I talked about the t-v distinction, which is the most common form of formal address in most languages. Italian, however, does things a little differently.
What does She want?
Instead of pluralizing the second person, or adding a title (like sir), Italians abstract into the third person. In the singular, this means that they call each other "she".
For example, when asking how someone is doing, if you are talking to someone with whom you are familiar you might say Tu, come stai? But when asking more formally, you would say Lei, come sta? Literally, you're asking "how is She doing?" Take note, however, that whether you are speaking to a man or a woman, you always use Lei as the formal singular subject pronoun.
In the plural, things are less weird, basically because there is no gender on the plural subject pronoun. But there is still a shift from 2nd to 3rd person. For example, when asking if a group of people are hungry, you might ask Voi, avete fame? But in the formal you would now say Loro, hanno fame?
When should you use the formal?
So when should the familiar form be used and when should the formal form be used? Here are some guidelines:
When talking to a family member, use the familiar.
When an adult is talking to a child, the familiar can be used.
When a child is talking to an adult, unless it is a family member, the formal should be used.
When young people talk to each other, the familiar is generally used.
When an employee is addressing a customer, the formal should be used.
With good friends or people you know well, the familiar form is used.
In almost all other cases, it is best to assume the formal should be used.
Using a familiar form where formal would be expected is considered disrespectful, so whenever you are in doubt, it is best to err on the side of formal.
Another style note is that English-speakers typically use "please" and "thank you" a lot as a way to show politeness, but because this can be established already by the tone of a request, such as by choice of suject pronoun, Italians use these courtesy words far more sparingly.
When people are getting to know each other, they start off on a formal footing, but after their relationship progresses, one will typically suggest Possiamo darci del tu? (Can we be familiar?) It would be rare to be refused.