Leaving aside ain't, as John suggests, the phenomenon called "double negation" is not so simple as it may seem.
Most of the languages of the world use multiple negatives to emphasize the negation, like the (respectively) French, Spanish, and Yiddish examples below:
- Je ne regrette rien. 'I don't regret anything' (lit 'I not regret nothing')
- No entiendo nada. 'I don't understand anything' (lit 'Not I_understand nothing')
- Ikh hob nit kin huyz. (איך האָב ניט קין הױז) 'I don't have a house' (lit 'I have not no house')
Using negatives this way is known as Negative Concord. Many English dialects, especially in informal registers, have a negative concord system; one well-known example is AAVE.
But other dialects, including standard whitebread American English, use a different system, called Negative Polarity. Instead of using negative elements in the focus of another negative to reinforce the negative sense, a negative polarity system uses other -- non-negative, but specialized -- elements, called Negative Polarity Items (NPIs), in these positions. From this article:
NPI is a term applied to lexical items, fixed phrases, or syntactic construction types that demonstrate unusual behavior around negation. NPIs might be words or phrases that occur only in negative-polarity contexts (ever, fathom, in weeks) or have an idiomatic sense in such contexts (not too bright, give a damn, drink a drop); or they might have a lexical affordance that only functions in such contexts (need/dare (not) reply); or a specific syntactic rule might be sensitive to negation, like Subject-Verb Inversion with Adverb Fronting in Never/**Ever*/**Frequently have I seen such a thing*.
And then there are several different types of over- and under-negation; there's a big literature on it.
Summary: Not all double negatives are incorrect;
and there's more to negation than you might expect.