1) These two conjunctions describe a situation where two subjects agree on a negative action. There are two forms. Be careful not to use a double negative: either a positive verb with the negative “neither" or a negative verb with the positive “either."
Statement + neither + Verb + Subject.
Statement + Subject + Negative Verb + either.
I don't like coffee, and neither does she. (not “neither doesn't she")
I don't like coffee, and she doesn't either. (not “she doesn't neither")
2) Be careful. They can also be used in positive sentences and even as the subject of a sentence.
Neither of the two proposals is satisfactory.
Neither is satisfactory.
Either of the two proposals is satisfactory.
Either is satisfactory.
3) They can be used as a positive or negative response to a question.
Would you like tea or coffee?
Neither, thank you.
Either is fine, thank you.
(Remember that “either" and “neither," by themselves, are always singular.)
4) And of course, there are the idiomatic forms.
Either + or
Neither + nor
Either the teacher or the students should know where to go.
Neither the teacher nor the students know where to go.
There is a unique structure for these typical idioms. Note that you will always use two nouns in this conjunction, and the number of the verb will always depend on the second example! Be careful.
Neither the ticket nor the credit cards are here.
Neither the credit cards nor the ticket is here.
Either the plane or the three buses are fine with me.
Either the three buses or the plane is fine with me.