In Italian there's the usage of "darsi del tu" (referring to your interlocutor in second person singular) and "darsi del lei" (using instead the feminine third person singular).
What's the difference? In what contexts should I prefer one form over the other? What is the origin of these forms? And are there other pronominal weridnesses I should be aware of?
It is also possible to use Voi instead of Lei. It is rare and used mostly in southern Italy. In some areas you are supposed to use "Voi" also speaking to your parents and also if you are not a kid anymore. This is very rare anyway, I met only 1 person following this rule.
Galateo states that you should use Tu with people of inferior rank, Lei with peers, Voi with people of higher rank and "Essi" with royal family member. Those rules are almost forgotten, it is very common to use Tu always except in formal or business situations as already answered.
By the way, using the family name to refer to a person is considered more formal and usually used with Lei, i.e.
Lei, signor Rossi, venga qua per favore! Tu, Mario, vieni qua per favore!
are proper sentences, instead
Lei, Mario, venga qua per favore! Tu, signor Rossi, vieni qua per favore!
are correct but they sound strange.
Italian has two forms of address meaning you in the singular: tu (the informal form) used with friends, family, children and animals; and lei (the formal form, normally written with uppercase "L"), which is used in business situations, such as in a shop or a bank, with new acquaintances (until invited to use tu) and with people older than you or considered worthy of respect.
There is also a formal form of address for more than one person (Loro) which a waiter or hotel receptionist may use when addressing more than one client.
None of the previous answers have addressed the origin of the forms "tu", "voi" and "lei", an aspect also asked by the OP and, for this reason, I would like to add some words.
According to Luca Serianni in his book Italiano (see section VII.85):
Il latino si serviva sempre del TŪ, qualunque fosse il livello dell'interlocutore. In età imperiale si diffuse il VŌS di rispetto che però, a quanto sembra (NICULESCU 1974: 12-15), non si è continuato nelle lingue romanze che avrebbero ricreato, autonomamente, un sistema oppositivo tu / voi. Nella Commedia Dante usa abitualmente il tu, riservando il voi «a persone per cui mostra il massimo rispetto» (ROHLFS 1966-1969: 477): per esempio Farinata degli Uberti, Brunetto Latini («Siete voi qui, ser Brunetto?» Inferno, XV 30), Cacciaguida («Voi siete il padre mio» Paradiso, XVI 16). Il lei si diffonde nell'uso cancelleresco e cortigiano del Rinascimento e si rafforza poi per effetto del modello spagnolo (MIGLIORINI 1957: 187-196).
That is, in Latin one used the form TŪ whatever was the level of the interlocutor. The form of respect VŌS spread in the Imperial Age, but it seems that this form hasn't continuity in Romance languages which developed an oppositive system tu / voi in an autonomous way. In the Divine Comedy, Dante usually uses "tu" and reserves "voi" to some specific characters to whom he shows maximum respect, such as Farinata degli Uberti, Brunetto Latini («Siete voi qui, ser Brunetto?» Inferno, XV 30) or Cacciaguida («Voi siete il padre mio» Paradiso, XVI 16). Lei form spreads in chancery and courtier uses during Renaissance and is then reinforced by influence of the Spanish model.
The origins of these Italian allocutive forms are addressed in detail in Lucie Kolková PhD thesis. About the origins of Lei, it is stated:
Il sistema bipartito stabilito nei primi secoli dopo Cristo fu arricchito nel periodo fra il Duecento ed il Quattrocento con un’altra forma allocutiva Lei che però si affermò in italiano solo nel Cinquecento.
That is, the bipartite system (tu / vos) established in the first centuries after Christ was enriched in the period between the thirteenth and fifteenth centuries with another allocutive form, Lei, but it wasn't consolidated in the in Italian language until the sixteenth century.
This confusion stems from the fact that English lacks what in Italian is called "forma di cortesia" nowadays (German has it, for example, and the Japanese one is more complicate, having different ways to express it depending on who the person you are addressing is).
As other answers already explained, "tu" is the normal way to address a person that you already know (friends, family members, colleagues) but with whom you have a strong relationship, or one that is reciprocated. This the normal use of this form, however there are instances in which a person of higher rank would say "tu" to a student, while expecting them to use "lei". While this can be normal and expected behavior, sometimes this difference is also used in a more rude way, e.g. when a native addresses an immigrant.
The other is used mostly with people that you dont't know or with whom you have a non-horizontal relationship, so to say, or often in business situations (when talking to customers, for example). Note that this ultimately comes down to respect.
One thing to bear in mind though is that these rules are not fixed in stone. They come from the Italian background and culture, and nobody is stopping you from using "tu" with a stranger, if you ask for permission to do so (otherwise it would be considered rude). So even if you are in a business situation, more often than not if you ask "possiamo darci del tu?", the other person will be more than willing to accept your offer.
Another thing to note when you are writing: usually you have to use a capital letter when you want to address somebody with "lei". So for example, if you are writing an email to ask for a piece of information, you would write:
Scrivo per chiederLe un informazione.
...
Approfitto per porgerLe
Cordiali Saluti
"Lei" is used in formal situations (at least until it is agreed by the parties that "tu" is more appropriate). I would normally consider it polite to wait for the "superior" (for lack of a better word) person to invite the other to go on "tu" terms, rather than for the "inferior" one to ask for it.
The English equivalent might be calling someone (named, say, John Smith) "Mr. Smith" rather than "John".
"Voi" is still very much used in the South, in particular by older generations.
