First of all there is special particle "ни-" in russian. If it's used as a distinct word - it means "neither". But if it's joined as prefix to word it becomes "no-".
-Где это было? - Нигде. (Where did it happen? -No-where)
-Кто это сделал? -Никто. (-Who did it? -Nobody (literally "no-who")
Moreover, there is "незачем" - no-why, something like "without reason".
This can explain next part - these words was born as answers to questions!
So, there is strange thing about ни-words - inside negatives they mean opposite things:
- Не пришёл никто - Anybody didn't come (literally, or "Nobody came")
- Я не был нигде - I didn't go anywhere.
Sometimes this form of speech is called "double negation", but it's strange to use such words in positive clauses. We don't say "Никто пришёл на работу" - it's wrong. English way to negate like in "Nobody wants to..." blows mind of russian learners of english first time they see it. "Ни-words" are always paired with negation or rejection.
So, imho, "ни-" does not form real double negatives (however some people thinks of it in this way) - it's just switchable nobody/anybody, depending on status of negation, special case of answer-words. I am not sure how teaching courses explain it.

Normally real "не" particle does only one thing: negates. And it negates only word after it, not anything else.
Я говорю плохие новости. - I am telling bad news.
Не я говорю плохие новости. - It's not me who is telling bad news.
Я не говорю плохие новости. - I am not telling bad news.
Я говорю не плохие новости. - I am telling not bad news.
Я говорю плохие не новости. - I am telling bad things which are not news.
You cannot put two "не" one after another - it just has no sense, so you cannot make double negation with it:
"Не я не говорю не плохие не новости" literally means "It's not me who is not telling not bad things which are not news". Gross! Nobody talks like this of course.

Maybe I miss something out.