Neither means ‘not one and not the other of two’. It is used before a singular noun.

Neither statement is true.

Neither of

Before a pronoun or a noun with a determiner (e.g. the, my, this etc.) we use neither of.

I like neither of them. (NOT I like neither them.) (NOT I like neither of they.)
Neither of my parents lives with me.

After neither of we use a plural noun or pronoun. The verb following neither of is singular in a formal style. It can be plural in an informal style.

Neither of my daughters speaks English. (Formal)
Neither of my daughters speak English. (Informal)

Neither and nor

As adverbs neither and nor mean ‘also not’. They come at the beginning of a clause and are followed by inverted word order.

I don’t like modern art. Neither do I. (NOT I also don’t.) (NOT Neither I do.)
Peter didn’t come, and neither did Sam. (NOT … and Sam didn’t too.) (NOT …and neither Sam did.)
I don’t like it, neither does she.

Not either has the same meaning as neither. It is used with normal word order.

I don’t like modern art. I don’t either.
I don’t like it. She doesn’t either.

Neither…nor

The correlative conjunction neither…nor is used to join two negative ideas.

He neither knows nor cares what happened.
Neither you nor I could have done this.

When neither…nor connects singular subjects, the verb is usually singular, but it can be plural in an informal style.

Neither Alice nor Mary sings well. (Normal)
Neither Alice nor Mary sing well. (Less careful)