In a similar way to how
ないでください means ‘please don’t do (A)’,
ないで means ‘without doing (A)’. This is a construction that combines the 助動詞（じょどうし） (auxiliary verb) ない, with the 格助詞（かくじょし） (case-marking particle),
で.
ないで usually has an (A), and a (B) clause, where the (B) clause will further explain another action that was done ‘without doing (A)’.
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日本（にほん）では靴（くつ）を脱（ぬ）がないで人（ひと）の家（いえ）に入（はい）ると怒（おこ）られます。
If you go into someone's house without taking off your shoes, they will get angry at you.
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晩（ばん）ご飯（はん）を食（た）べないで待（ま）っててくれたの？ごめんね。
You waited for me without eating dinner? I'm sorry.
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カーテンを閉（し）めないで寝（ね）たから朝（あさ）の4時（じ）に起（お）きた。
I slept without closing the curtain, so I woke up at 4 in the morning.
When using
ないで, the (A) phrase is often something that is ‘expected’ to have been done, and is therefore surprising in some way that (B) was accomplished without it.
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昨日（きのう）は風呂（ふろ）に入（はい）らないで寝（ね）た。
Yesterday, I went to bed without taking a shower. (Despite having a shower being expected before bed)
Be careful of the following two structures:
(A)
ないで (B) - Without doing (A), (B)
(A)
なくて (B) - Without doing (A), (B)
Technically, these two constructions share a similar meaning. However, in modern Japanese
ないで is used more frequently when the (B) action is being highlighted as being done without (A). On the other hand,
なくて is used when (A) is being given as a reason for why something else was/wasn’t done.
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昼（ひる）ご飯（はん）を温（あたた）めないで食（た）べた。
I ate my lunch without heating it up. (Event 2 happened without event 1)
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インターネットが繋（つな）がらなくて困（こま）ってる。
I am bothered because the internet isn’t connecting. (Event 2 happened because of event 1)