lei is the courtesy pronoun in Italian. Similar in concept to the Spanish usted, to the French vous, to the Dutch U and to countless other courtesy pronouns.
lei is in fact the feminine indirect pronoun third person singular, more or less like her in English only different.
Italians are so accustomed to lei that they see it like this any more, but in fact when you are talking to someone with lei you are speaking as if with an invisible woman that happens to be in the same room.
Le da fastidio se fumo?
literally: Does she mind if I smoke
actually: Do you, Sir or Madam, mind if I smoke.
La sua automobile è parcheggiata sul mio piede
literally: Her car is parked on my foot
actually: Your car is parked on my foot.
And if you think that this sounds strange, I'll tell you that another, quite old fashioned, courtesy form uses the voi (the Fascists tryed to impose this), that's to say the second person plural: in which you pretend that you are talking with a whole bunch of people.