Neither and nor can be used as adverbs to mean ‘also not’. Note that neither and nor come at the beginning of a clause, and are followed by inverted word order: auxiliary verb + subject.

I can’t dance. Neither can I. (NOT I also can’t.)
Mary didn’t come. Neither did James. (NOT James didn’t too.)
Alice wasn’t invited. Neither was Mary.

Not either can be used with the same meaning and normal word order.

She can’t sing. Her brother can’t either. (= Neither can her brother.)
I don’t like science fiction. Neither do I. OR I don’t either.

In a very informal style, me neither can be used instead of ‘I can’t either’.

I can’t swim. Me neither. (= Neither can I.)

Not…nor

In American English, nor is not used after and.

John didn’t turn up, and neither did Peter. (US)

John didn’t turn up, and nor / neither did Peter. (GB)

Nor can follow not; it is more emphatic than or.

Our priority is not money, nor fame. It is education. (More emphatic than ‘Our priority is not money, or fame’.)