In Italian, a polite form of address for someone that holds a university degree. At what level, you might ask. Since the Italian system used to have only one degree, the Laurea, this used to be a non-question. Recently, with the move of the Italian higher education system towards a more European model, people have extended Dottore (shortened by apocope as Dottor) to shorter degrees. In any case, keep in mind that Dottore does not imply a PhD!
When should it be used? Before a surname when being quite formal: "Buongiorno, Dottor Rossi" "Come sta, Dottor Zamberletti?".
The feminine form is Dottoressa "Si accomodi, Dottoressa Bianchi!"
Dottor can also be used by itself (much like "Doctor" in English).
In some circles, beyond a certain management level everybody is called Dottore; or, amusingly, nobody is - because it is assumed that everybody has a degree.
I have also heard it used rather pointedly in university circles, to remind people of the fact that a certain person was not a professor. Some Italians believe that being called Dottore before you graduate leads to the fact that you will never graduate - so never do that.