Double negatives can be perfectly fine in English.

- If their sum is supposed to be **negative**, double negatives are very informal or slang in modern English. This usage is frowned upon by many people even if used in speech, unless ironically.

> 1.) I **don't** see **nothing**. *(= I don't see anything.)*

- If their sum is supposed to be **positive,** it is generally acceptable in all registers:


> 2.) I suppose that is **not impossible**. However, it seems far fetched.

Here the double negative expresses a weak positive, a very common construction.

> 3.) **Not bad, not bad** at all! You have just saved her life, young man.

This is a figure of speech called *[litotes][1]*: the double negative is used to express a strong positive. Sometimes any double negative with a positive meaning is considered a litotes, including the unremarkable example 2 above. Other people restrict this term to those negations that express a strong positive through an apparently weak positive, in a mildly ironical manner, as in this example (3). 

> 4.) **Never** a day goes by that I do **not** miss her. *(= I always miss her.)*
 
This is the rhetorical double negative, often considered a form of litotes. It expresses a strong positive, though without irony.

> 5.) Well, I **didn't *not*** enjoy it, but...

Here the word *not* is used twice, once in contracted form (*don't*) and once in full, to express a weak positive. Double *not* is a special case: it is felt to be even more redundant than other double negatives and sounds rather colloquial. The majority will probably use this in speech and informal writing, where it is perfectly acceptable, but not elsewhere.

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The boundary between negations and other kinds of words is by no means sharp. The prefixes *un-* and *im-/in-* are usually considered negatives, as are *hardly* and many others; *bad* is sometimes considered a negative word, sometimes not, etc.

  [1]: http://en.wikipedia.org/wiki/Litotes