Although it is used almost exclusively as a 接尾語（せつびご） (suffix), なし is actually a noun that means ないこと ‘thing without’. なし, or 無（な）し when using kanji, will be used after nouns that it is indicating as ‘not having (A)'.
Depending on what the speaker wants to convey, なし is often partnered with the 助動詞（じょどうし） (auxiliary verb) だ, or its conjunctive form で. なしで will mean ‘(A) that is without (B)’. When connected directly to another noun, の will be required after なし. Again, this will mean ‘(A) that is without (B)'.

• その考（かんが）えは間違（まちがい）いなしだ。
  That way of thinking is without error. (That way of thinking is flawless)
• 許可（きょか）なしで公園（こうえん）に店（みせ）を出（だ）さないで下（くだ）さい。
  Please don't put up a store in the park without permission.
• 肉（にく）なしの人気（にんき）料理（りょうり）はなんですか。
  What is a popular dish without meat?

なし does not always need to be connected directly to another noun. Often, it can be used in reference to an event or thing that someone wants to skip, or avoid. In cases like this, it may be paired with the 格助詞（かくじょし） (case marking particle) に, to show determination to avoid something.

• 昨日（きのう）は沢山（たくさん）食（た）べたから、今日（きょう）は昼飯（ひるめし）なしにする。
  Because I ate a lot yesterday, I am going to skip eating lunch today.
• 先輩（せんぱい）が来（く）ると盛（も）り下（さ）がるから、先輩（せんぱい）なしにしよう。
  Let's avoid inviting our senpai because he kills the mood.