We use 'either + a singular noun' to mean 'this one or that one' when we are talking about two things of the same kind (like two drinks or two t-shirts).
Either drink is fine. I like both kinds.
We could go to either restaurant.
Julie might buy either t-shirt.
We use 'neither + a singular noun'' to mean 'not this one and also not that one' when we are talking about two things of the same kind.
Neither drink is fine. John hates both of them.
Neither restaurant is good. Let's go somewhere else.
We use 'either of + plural noun' and 'neither of + plural noun' before a pronoun or a word like 'this' or 'the' or 'my'.
Either of my dresses is good. (= this dress is good and also that dress is good.)
Either of us can get the key.
Neither of my dresses will do. (both of my dresses are not good.)
Neither of us will be on time.
We can use 'either' and 'neither' on their own when we have already said the noun.
Which drink would you like? Either is fine.
Which cake do you want? Neither! I'm going to have an apple.
If we want to talk about two different things, we can use 'either + noun + or + noun'. We don't need to use 'of' in this case.
Either coffee or tea is fine.
We could use either blue paint or red paint.
She will buy either some apples or some oranges.
We can also use 'neither + noun + nor + noun'. This tells us about two negative ideas, but it's quite formal. It's more normal just to use 'not + or'.
I don't like coffee or tea.
I like neither coffee nor tea.
We can also use 'neither' at the beginning of a sentence or clause. It means 'also not'. We need to use inverted word order afterwards.
I'm not at home. Neither is John.
This is often used to answer someone.
A: I don't like coffee. B: Neither do I.
In an informal style we can use 'me neither'.
A: I don't like coffee. B: Me neither.
We use 'not... either' to mean the same thing, but we use normal word order.
A: I don't like coffee. B: I don't either.
Here´s some practice on the use of either and neither: