It is the use of either and neither (rather than the spelling) that people seem to have trouble with – I have just edited a 40 page document which repeatedly misused either/or.

either: an introduction of two alternatives separated by the word or
You can either buy or rent this property

neither: an introduction of two unavailable alternatives separated by the word nor
neither John nor Mary can run the meeting.

What is important to note is that both words refer to a choice of TWO options – if there is a list of choices, either/neither can’t be used.

wrong – You can choose either chocolate or vanilla or strawberry.
correct – You can chose either chocolate or strawberry.
correct – You can choose chocolate, vanilla or strawberry.