Quick answer
не goes in front of the thing you are denying: Я не зна́ю. нет means “there is no” and takes the genitive: У меня́ нет вре́мени.
не — the everyday no
Я не зна́ю.I don’t know.
Он не до́ма.He is not at home.
Не я, а он.Not me — him. — не goes before the word it denies
нет — there is no
Нет, спаси́бо.No, thank you.
У меня́ нет вре́мени.I have no time. — + genitive
Здесь нет метро́.There is no metro here.
Double negatives are correct
Where English allows only one negative, Russian requires two — the negative word and не together.
Я ничего́ не зна́ю.I don’t know anything. — literally “I nothing don’t know”
Никто́ не пришёл.Nobody came.
Я никогда́ не был в Москве́.I have never been to Moscow.
The rule in one line
If the sentence contains ничего́, никто́, никогда́, нигде́, it must also contain не. Leaving it out is the mistake, not the fix.
Typical mistakes
Я ничего́ зна́ю.
Я ничего́ не зна́ю.
The negative word needs не with it.
У меня́ не вре́мя.
У меня́ нет вре́мени.
Absence is нет plus the genitive.
Check yourself
How do you say “I never drink coffee”?
никогда́ needs не in front of the verb — two negatives, one meaning.