〜ずに
without doing
Verb ない-form minus ない + ずに (する → せずに)
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ずに means "without doing", like ないで: 朝ご飯を食べずに出かけた, "I went out without eating breakfast". Build it by dropping ない from the negative and adding ずに: 食べない → 食べずに, 行かない → 行かずに.
The one exception is する, which becomes せずに, not しずに: 勉強せずにテストを受けた, "I took the test without studying".
ずに is a little more formal and written than ないで, so you'll see it in books, news and instructions. In conversation ないで is more common, but the meaning is the same.
ずに only means "without"; it can't make a request, so 食べずにください is wrong. Use 食べないでください for that.
Examples
朝ご飯を食べずに学校に行きました。
I went to school without eating breakfast.
辞書を使わずに読めますか。
Can you read it without using a dictionary?
昨日は勉強せずに寝てしまった。
Yesterday I went to sleep without studying.
傘を持たずに出かけた。
I went out without taking an umbrella.
誰にも言わずに会社を辞めた。
They quit their job without telling anyone.