ないで is a te-form of the negative form of a verb. There are basically two different -te forms, ~ないで and ~なくて, for the nagative form "verb + ない", whether the verb is an irregular or godan/ichidan. You can think expressions like ~ないでください, ~ないでほしい, ~なくてはいけない, ~なくてはならない, ~なくてもいい are a kind of set phrase, i.e., ~なくてください can't be used, for instance. (The soft command ~ないで is considered a variation of ~ないでください.) Generally, ~ないで is more widely used, and ~なくて is often used to express "cause/reason" (e.g. 彼がいなくて困った "I was in trouble since he wasn't there").