〜ないで
Doing B without doing A; don'tmachine translation
FormationVerb ない-form + で
〜ないで links two actions by the same person, where A doesn't happen while B takes place: 朝ご飯を食べないで学校に行きました (I went to school without eating breakfast). The main action is the second clause; the 〜ないで clause just describes the accompanying state.
You met 〜ないでください at N5 meaning "please don't". In casual conversation, Japanese speakers drop ください and just say 〜ないで: 行かないで! (don't go!).
Don't confuse it with 〜なくて: 〜ないで means "without doing" (an accompanying state), while 〜なくて gives a reason, "because … not …". Compare: 食べないで出かけた (went out without eating) and 食べられなくて残念だった (it was a shame because I couldn't eat it). Also, 〜ないで is used only with verbs.
Examples
朝ご飯を食べないで学校に行きました。
I went to school without eating breakfast.
辞書を使わないで新聞を読みます。
I read the newspaper without using a dictionary.
電気を消さないで寝てしまいました。
I accidentally fell asleep without turning off the light.