なさい is the 命令形（めいれいけい） (imperative form) of なさる, a 尊敬語（そんけいご） (honorific language) verb which specifically highlights and respects the actions of another person (never the speaker).
Despite being respectful, the fact that this is a ‘command’ means that it would never be used toward a superior. Due to this, なさい is most frequently used toward peers, or people of lower status, like children.
To use なさい, simply connect it to the ます stem of any verb.

• 早（はや）く寝（ね）なさい！
  Hurry up and sleep!
• 遊（あそ）んでいないで勉強（べんきょう）しなさい！
  Stop playing and study!

なさい may be shortened further to な, which becomes an even more casual form that is used most frequently toward peers.

• 勉強（べんきょう）しな。
  You should study.
• お前（まえ）も悪（わる）いから謝（あやま）りな。
  You are also at fault, so you should apologize.