Nor means 'and not'. It is usually a mistake, in the eyes of pedants, to say 'and nor'. This properly means 'and and not', and is therefore an unnecessary repetition.
(Nor became popular in English, where its use is not particularly natural, to represent a connecting word in Latin, in the days when education was most concerned with the classics. In texts of that era, nor is sometimes used to begin a negative sentence. This echoes the common start to a Latin sentence, Nec.)
Nor is still the appropriate word to follow neither, as or is after either. Note, however, that while either/or offers a choice, neither/nor, since it excludes both alternatives, does not offer a choice. (See further neither and nor.)
In the fields of electronics, formal logic and computer programming nor is a contraction of the operator 'not or'.