〜ないで
without doing, instead of doing
Verb ない-form + で
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ない-form + で means "without doing" or "instead of doing": 朝ご飯を食べないで学校に行きました, "I went to school without eating breakfast".
It's the negative partner of the て-form that joins actions: 食べて行った (ate, then went) vs 食べないで行った (went without eating).
It's the same ないで as in ないでください ("please don't"), which is really "please (go on) without doing".
Don't mix it up with なくて, which gives a reason: 分からなくて困った, "I didn't understand, so I was stuck". A rough test: if English says "without", it's ないで.
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.
傘を持たないで出かけた。
I went out without an umbrella.
辞書を使わないで読んでみてください。
Try reading it without using a dictionary.
昨日は寝ないで勉強しました。
Yesterday I studied all night without sleeping.
電車に乗らないで、歩いて帰りました。
Instead of taking the train, I walked home.