ないでください is an expression that is used to politely ask someone ‘not’ to do something. It is a combination of the casual negative form of a verb, the で particle, and ください (the conjunctive form of くださる).

• これは誰（だれ）にも言（い）わないでください。
  Please don't tell this to anyone.
• そこには座（すわ）らないでください。
  Please don't sit there.

ください is classified as 尊敬語（そんけいご）, a type of polite speech that is used to refer to the actions of other people. The ください form of くださる is used almost exclusively for making requests. くださる is the polite (sonkeigo) equivalent of くれる.
In friendly conversation, ください may be omitted from this expression, and the で form itself can be used as a very casual way of saying ‘please don’t do (A)’.

• それ、お兄（にい）ちゃんのだから、食（た）べないで。
  Because that's my older brother's, please don't eat it.
• 後（あと）で電話（でんわ）するから寝（ね）ないでよ。
  Because I'm going to call you later, please don't sleep.