Russian, unlike English, has two different forms of the second person singular pronoun "you": вы and ты. Compare these to the French Vous and tu or the German Sie and du. The вы form is also the second person plural.
Вы is used in formal situations, work settings, when speaking to a superior or to someone older, to whom you wish to show a degree of respect. Вы indicates distance (emotional, social or chronological).
Ты is used in informal situations, family relationships (between spouses, parents and children, between siblings and with other relatives), when speaking to someone younger than you. Ты indicates closeness (in age or in terms of friendship and intimacy).
Вы-forms of address can be full first name plus patronymic, full first name minus patronymic, or shortened first name only. Ты-forms of address will be first name only, either full or shortened.
Ты is always used when addressing God, speaking to children (your own or anyone's) and calling your cat or dog.
When they first meet, adults will usually begin by addressing each other as Вы. At some point in time in the relationship, one may suggest to the other that they change to ты by saying: Дава́йте перейдём на ты; ог Мо́жно на ты? If this is mutually agreeable, the relationship passes onto a new level; some people even make a ritual of it by drinking "брудерша́фт".