Polite and familiar address in Russian
Russian grammar --> Polite and familiar address
The Russian language has two forms of address: familiar ты and respectful Вы. The polite form is used when you talking to someone you don't know very well or in formal situations. Russian familiar form ты is generally used among friends and relatives.
People may transition from ты to Вы with mutual agreement and begin to speak on familiar terms. Use of ты without consent of the other person is considered to be extremely rude.
Grammatically, Russian polite form Вы is identical to plural of second person вы (note that the polite form is capitalized and plural is not), so other related sentence elements like verbs and adjectives require the usage of their plural forms.
Examples:
Я надеюсь, ты понимаешь. (I hope you understand) - familiar form of address to one person, verb понимать (to understand) in singular of second person.
Я надеюсь, Вы понимаете. (I hope you understand) - respectful form of address to one person, verb понимать (to understand) in plural of second person.
Я надеюсь, вы понимаете. (I hope you understand) - address (respectful or familiar) to several persons , verb понимать (to understand) in plural of second person.