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• I ate simple positive statement
• I didn't eat simple negative statement
• I didn't eat anything negative statement
• I didn't eat nothing double negative which means "I ate something". This is a very rare construction which which emphasises that it isn't nothing. "I didn't eat nothing, but it was very small."
• I didn't not eat double negative, which means "I ate". This is a very rare construction which emphasises that it happened. "I didn't not eat, but it was so small it wasn't really a meal.

This is a double negative, which is absolutely correct in Standard English but is rare. Its meaning is double: a) "you go nowhere", and b) I don't want that. This gives a positive sense, but is subtly different from the positive statement "I want you to go somewhere", and would only be used in sentences such as this: "I do not want you to go nowhere, I want you to go somewhere." As noted in comments, there is emphasis on the nowhere. (Thanks @chrylis) It is often needed to say things like "It's isn't zero, but it is very small." One of the most common uses of correct double negatives is in an argument: Bert: "I did everything. You did nothing." Ernie: "I didn't do nothing, I did my share."

Negative Concord in Nonstandard English

There are many varieties of non-standard English which have "negative agreement", also called "negative concord", and you will hear "I didn't eat nothing" where standard English has "I didn't eat anything". Many languages have negative concord, such as Russian, Spanish, Hebrew, and the general idea is that the parts of a sentence which are negatable must all agree. Just as in English we have gender agreement -- "She said she herself is tall like her father" must become "He said he himself is tall like his father" -- some grammars require all the variable parts of the sentence to match in positive or negative forms.

• I ate simple positive statement
• I didn't eat simple negative statement
• I didn't eat anything negative statement
• I didn't eat nothing double negative which means "I ate something". This is a very rare construction which which emphasises that it is something, not nothing. "I didn't eat nothing, but it was very small."
• I didn't not eat double negative, which means "I ate". This is a very rare construction which emphasises that it happened. "I didn't not eat, but it was so small it wasn't really a meal.

This is a double negative, which is absolutely correct in Standard English but is rare. Its meaning is double: a) "you go nowhere", and b) I don't want that. This gives a positive sense, but is subtly different from the positive statement "I want you to go somewhere", and would only be used in sentences such as this: "I do not want you to go nowhere, I want you to go somewhere." As noted in comments, there is emphasis on the nowhere. (Thanks @chrylis) It is often needed to say things like "It isn't zero, but it is very small." One of the most common uses of correct double negatives is in an argument: Bert: "I did everything. You did nothing." Ernie: "I didn't do nothing, I did my share."

Negative Concord in Non-Standard English

There are many varieties of non-standard English which have "negative agreement", also called "negative concord", and you will hear "I didn't eat nothing" where standard English has "I didn't eat anything". Many languages have negative concord, such as Russian, Spanish, Hebrew, and the general idea is that the parts of a sentence which are negatable must all agree. Just as in English we have gender agreement -- "She said she herself is tall like her father" must become "He said he himself is tall like his father" -- some grammars require all the variable parts of the sentence to match in positive or negative forms. In Spanish when you negate "Tengo sed también" (literally "I have thirst too") it becomes "No tengo sed tampoco" (literally "Not I have thirst neither"): we add "no" and alter "tambien" into "tampoco".

Standard English, as you will read on the BBC and the New York Times, has "negative polarity", which means that a double negative is actually a positive, though not necessarily with the identical intention as the obvious positive statement. A "double negative" is where there are two negatives in the construction, although many people use the phrase only to mean a "double-negative error".

There are many varieties of non-standard English which have "negative agreement", also called "negative concord", and you will hear "I didn't eat nothing" where standard English has "I didn't eat anything".

Standard English, as you will read on the BBC and the New York Times, has "negative polarity", which means that a double negative is actually a positive, though not necessarily with the identical intention as the obvious positive statement. A "double negative" is where there are two negatives in the phrase, although many people use the phrase only to mean a "double-negative error".

There are many varieties of non-standard English which have "negative agreement", also called "negative concord", and you will hear "I didn't eat nothing" where standard English has "I didn't eat anything". Many languages have negative concord, such as Russian, Spanish, Hebrew, and the general idea is that the parts of a sentence which are negatable must all agree. Just as in English we have gender agreement -- "She said she herself is tall like her father" must become "He said he himself is tall like his father" -- some grammars require all the variable parts of the sentence to match in positive or negative forms.

Direct answer

Standard English, as you will read on the BBC and the New York Times, has "negative polarity", which means that a double negative is actually a positive. A "double negative" is where there are two negatives in the construction, although many people use the phrase only to mean a "double-negative error".

This answer is long because it can be very difficult for learners to distinguish a) Correct double negative in Standard English, b) Double negative error in Standard English, and c) Correct negative concord in non-standard English.

Direct answer

Standard English, as you will read on the BBC and the New York Times, has "negative polarity", which means that a double negative is actually a positive, though not necessarily with the identical intention as the obvious positive statement. A "double negative" is where there are two negatives in the construction, although many people use the phrase only to mean a "double-negative error".