The 'Golden' Rule of the Double Negative

As early as your first English lessons, the teacher says: "English does not have a double negative. If we use two, they cancel each other out and the sentence becomes positive. So if I say „I couldn’t not help him.”, that actually means I helped him."
And then, right after the teacher declares that the double negative is not a negative, we put on some music and hear Pink Floyd sing "we don’t need no education", and that Rolling Stones „can’t get no satisfaction”. Why is this possible and who is right, the teacher or Pink Floyd?

In short, the double negative becomes a positive. In the spoken language, however, especially in the USA, it is sometimes used as a negative, usually in some phrases. I would not advise learners to use it. On one hand it is to be avoided in writing, and on the other hand you really want to avoid misunderstanding, so stick to the grammatically correct form.

Using a double negative is indeed correct when we are trying to express a positive thought, in other words do what the teacher told us to do: neutralize one negative with the other, for example:
You can’t not talk to anybody OR You can’t talk to nobody.
When I look back I don’t regret not going to school.
We can’t just do nothing when we are feeling threatened.

Which words are considered negative and used together form a double negative: no, not, no one, none, nobody, neither, nowhere, never, nothing. Of course, this includes the contracted „not”, as in hasn't, shouldn't, couldn't.

You need to be careful with adverbs like hardly, barely, scarcely, seldom and rarely, which carry a negative meaning and do not need an additional "not":
He couldn’t hardly breathe.
√ He could hardly breathe.
They have barely no money.
√ They have barely any money.

Sample phrases, used with the double negative in English, although grammatically incorrect are for example:
I ain't got no ... (I don't have any ...)
never ... no one. (never … anyone), for example: Never trust no-one!, A little bit of fun never hurt no one.
I don't never ... (I never ...)
don’t know nothing about ... (don't know anything about ...)
It won’t do you no good. (It will do you no good.)