Introducing Yourself in Russian
When you meet someone, the first thing you do is say hello and give your name. In Russian, the most common way to say your name is with the verb звать (zvat', "to call"). Literally you say "they call me...". So Меня зовут Анна (Menya zovut Anna) means "My name is Anna."
To ask someone else's name, you say Как вас зовут? (Kak vas zovut?) — "What is your name?" (formal/polite). With friends or children you can use the informal Как тебя зовут? (Kak tebya zovut?).
Nice to meet you
After exchanging names, say Очень приятно (Ochen' priyatno) — literally "very pleasant", meaning "Nice to meet you." You can also introduce yourself simply with Я (Ya, "I") plus your name: Я Иван (Ya Ivan) — "I am Ivan."
Key words
Меня зовут (Menya zovut) — My name is
Как вас зовут? (Kak vas zovut?) — What is your name? (polite)
Очень приятно (Ochen' priyatno) — Nice to meet you
Я (Ya) — I / I am
Привет (Privet) — Hi / Hello (informal)
Examples
Привет! Меня зовут Мария. (Privet! Menya zovut Mariya.) — Hi! My name is Maria.
Как вас зовут? — Меня зовут Пётр. (Kak vas zovut? — Menya zovut Pyotr.) — What is your name? — My name is Pyotr.
Очень приятно! (Ochen' priyatno!) — Nice to meet you!