When used as a determiner, either and neither must come before a singular countable noun.
Either is positive (+). It means "one OR the other" (of two).
- A. Do you want to meet on Monday or Tuesday?
B. Either day is good for me. (Monday is good for me. Tuesday is good for me.)
- A. There are two keys here. Which will open the office?
B. I believe either key works.
Neither is negative (-). It means "not one AND not the other" (of two).
- Neither team scored and the game was a draw. (Team A didn't score. Team B didn't score.)
- Neither player was allowed to return to the game after the argument.
either: /'i:ðə/ OR /'ʌɪðə/
neither: /'ni:ðə/ OR /'nʌɪðə/
Either pronunciation is acceptabe. Neither pronunciation is wrong.
More information on specific quantifiers: