Sometimes you can use either one and sometimes you have to choose either one or the other, but neither one is very difficult. ( understood? )
Ok… let’s explain:
Either and Neither are used with an auxiliary or modal verb to express agreement in the negative
Either… or
Either… or is used to offer a choice between two possibilities:
Either Mike or Lisa will be there.
Either you leave me alone or I will call the police.
We should bring either coffee or tea.
You can either help us or go to your room.
Either can also be followed by (one) of + group of two:
Either of us could do it
Either one of us could do it
Either of you should know
Either one of you should know
Not… either… or denies both possibilities:
I don’t think either Mike or Lisa will be there.
He doesn’t speak either English or French.
Not… either is used after a negative statement.
I don’t speak French.
You don’t either.
He isn’t ready to go.
We aren’t either.
Neither… nor
Neither… nor is equivalent to not… either / either… or.
Neither Mike nor Lisa will be there.
He speaks neither English nor French.
We brought neither coffee nor tea.
I will neither help you nor go to my room.
Neither can also be followed by (one) of + group of two:
Neither of them are ready.
Neither one of them are ready.
Neither of us has any money.
Neither one of us has any money.
Neither is used like not… either.
I don’t speak French.
Neither do I. / I don’t either.
He isn’t ready to go.
Neither are we. / We aren’t either.
The bottom line is that,
Either means one, neither means none,
not either = neither.
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If you wish to take a closer look, check The American Heritage® Book of English Usage topics about Either and Neither.