It's taken me awhile to get around to this subject, not because it's particularly difficult (it isn't), but because it reflects one of the aspects I like least about Italian. I'll explain this a bit later.
As you probably know, you use the informal tu with friends, family, and children, and the formal lei as the default. Those of you who have actually tried this out with Italians know it's not that simple. For example, if you are middle-aged and you are speaking to a young adult, it may well happen that they address you with lei, and you can continue on your merry way with tu. Among people in the same age group, the tu often comes up spontaneously (rarely among the elderly, though.) Using the tu with someone you don't know may also be a sign of affection, of feeling close to them. Online communications foster this. Another ostensible exception: in prayers the Italians use the tu to speak to God (Padre nostro, che sei nei cieli) and the Virgin Mary (Ave Maria, piena di grazia, il Signore e' con te). Maybe they're perceived as family members. However, if you try addressing the Pope with the tu the next time you run into him, you'll be in a heap of trouble. And don't think that just because he's German he won't notice. He, and everybody else, will be scandalized.
Speaking of German and some other languages. They, too, have their formal and informal pronouns. Do not use these as a guide to Italian usage, even with languages that are similar to Italian, such as French. The French use of tu and vous is not the exact same as Italian usage of tu and lei.
Voi used to be an alternative to the other two, proposed, along with other silly linguistic reforms, by one Benito Mussolini. You can of course use voi when two or more people are present, and this gets you out of the choosing dilemma. In textbooks and sites that are not with it, they will have loro as the plural equivalent of lei . This is overkill, scrap it. I have only ever heard loro in Italy used this way by stuffy waiters. Also, avoid capitalizing the Lei. This is also rapidly on its way out.
The main problem with these forms is not the usage, but that you have to change your grammar along with it. In other words, you have to use the third person instead of the second. Thus, if you wished to indicate to a stranger that they can precede you, you would say "vada," not "va'." Or to ask someone to follow you: "venga," not "vieni."
An unusual but potentially comical situation sometimes arises when the formal lei and the feminine lei are confused in conversation. People may be referring to you, and you say "Lei chi?" as if they were talking about some woman who is not present.
Lei is used not only as a sign of respect but also as a way to keep your distance. The Italians, in passing from the formal to the informal, will usually ask permission by saying possiamo darci del tu? The answer is usually "yes." Why do I dislike this feature of Italian? It forces you (and not only you as a non-native) to constantly choose between the two forms, and does not seem to befit a democratic people. I much prefer our English here. It even allows people to subtly insult you by inappropriately using the tu. This is done not rarely, and you will sometimes hear indignant Italians say: "mi dia del lei!" "Use the lei form with me!"
You, as a furriner, may well have a mal di testa by now. You may even have fallen into a depressione profonda. Be of good cheer! Italians know this isn't the easiest thing for non-natives, so they cut you a lot of slack. They will also sometimes use the tu with foreigners (especially if your knowledge of Italian is scarce or non-existent) to make things easier for you, and not to put you down.
