Use either when you talk about two things or people.
Examples:
(1) “I would like either cookies or popcorn.
I can’t decide.”
(2) “I would like either (of the snacks).”
(3) “I don’t want to eat either snack.”

Use neither (for negative sentences) when you talk about two things or people. Examples:
(1) “Jack has been to neither Canada nor the United States.”
(2) “Jack has been to neither (of the countries).”
(3) There is no negative for neither.

Grammar Note:
(From the American Copy Editors Society)
A tricky agreement problem: Neither-nor
When two subjects are joined by neither-nor or either-or, choosing the right
number for the verb can be tricky for writers. Focus your attention on the noun
closest to the verb. If it is singular, as in the sentence above, choose the singular
verb. If the noun is plural, choose the plural form of the verb.