Which is correct: Anyway or Anyways

Anyway is correct. "Anyways" is not a word, but it is used out of ignorance by many.

Many properly educated speakers find "anyways" to be grating on the ear and will think less of the speaker who misuses "anyway" by adding the "s". The Grammar Girl website is full of examples where job applicants were turned away because they used "anyways".

Just because you hear it on television, doesn't make it right or acceptable. Television will often use bad grammar as a way of telling us about a character and her background. Please avoid using "anyways".

Edit: I've added to this answer to elaborate on Rahul's point.

The Oxford Dictionaries take a descriptivist view of the language. They try to describe how the language is being used, not whether that use is right or wrong. Other dictionaries -- notably Funk and Wagnall's -- take a more prescriptive view and only include words/usages they believe to be correct. My answer leans heavily toward the prescriptive camp.

Garner's Modern American Usage (3rd edition, 2009, Oxford University Press) tries to take a middle ground. Where there is disagreement about a word's suitability, Garner's provides a five-step scale ranging from outright rejection to full acceptance. Anyways receives a 3 from Garner's, meaning that while it is widely used informally, it remains substandard and should be avoided by discerning writers and speakers.

It is well said that 'anyways' is a derivative of "Anyway". It is used informally, as Oxford Online Dictionary suggests.

See here:

http://oxforddictionaries.com/definition/english/anyways?q=anyways

I have edited by original answer to elaborate on Rahul's answer.