“Either” and “neither” are used in sentences concerning a possible choice between two items where the person using the word(s) is not concerned about the differences between the two items [ either] or where they decide not to make a choice [neither]
-It is used to reject 2 possibilities (not more. For more, you use none, any) e.g. I have 2 dresses, but neither is appropriate for the cocktail party.
-It is used to agree with a negative statement. e.g. I don’t want any coffee – Neither do I.
-It is used (combined with “or”) to propose 2 possibilities e.g. I’ll visit my aunt either on Monday, or on Tuesday.
-It is used to agree with a negative statement (as neither) BUT only if you have a negative phrase e.g. I don’t like this dress! – I don’t like it, either.(or “neither do I”, it’s the same thing)