Quick answer
О́чень прия́тно!
óchen priyátno — literally “very pleasant”
The options
О́чень прия́тно!Nice to meet you!
Прия́тно познако́миться.Pleased to meet you. — a little fuller
Рад познако́миться.Glad to meet you. — a man
Ра́да познако́миться.Glad to meet you. — a woman
Мне то́же.You too.
Who says it first
The person being introduced usually says it first, and the other answers Мне то́же or simply repeats О́чень прия́тно. A handshake goes with it between men; between men and women it is optional and usually initiated by her.
What you hear back
Мне то́же о́чень прия́тно.Nice to meet you too.
Взаи́мно.Likewise.
Приве́т!Hi!
Typical mistakes
Хорошо́ встре́тить вас.
О́чень прия́тно.
A word-for-word translation of the English. Russians use the fixed phrase.
Рад — said by a woman
Ра́да
Short adjectives agree with gender.
Check yourself
A woman says she is glad to meet you. Which form does she use?
Feminine short form: ра́да.