We can use either or neither as a pronoun, determiner, conjunction or an adverb.
1. Used as a Pronoun
EXAMPLE: Either of the boys is guilty of starting the fire.
EXAMPLE: Neither of the two donkeys carries heavy loads.
EXAMPLE: Either of them has stolen my chewing gum.
EXAMPLE: Neither of them has the courage to talk to her.
EXAMPLE: Give me either of them please.
EXAMPLE: Neither of the boys knows anything about her.
EXAMPLE: "Which pizza do you want?" "Neither, Thank you." (There are only two pizzas.)
v Instead of using both ... followed by not ... we would use neither.
EXAMPLE: We would not usually say, "Both of them don't know the answer." Instead, we would say "Neither of them knows the answer."
v We can use either or neither on its own.
EXAMPLE: "Do you want coffee or tea?" "Either. It doesn't matter which one."
EXAMPLE: "Which house do you have in mind?" "Neither. They are overpriced."
3. Used as a Conjunction
EXAMPLE: You can borrow either one or two books for now.
EXAMPLE: Either she comes over to my place or I go over to hers as agreed.
EXAMPLE: It's either you send your old fridge for repair or you buy a new one.
EXAMPLE: Neither the cat nor the dog is black in colour. (NOT 'are'.)
EXAMPLE: Either this frog or that toad is croaking. (NOT 'are')
4. Used as an Adverb
EXAMPLE: She won't come with me or with him either.
EXAMPLE: "I can't spell very well." "I can't either." or "Me, either."
EXAMPLE: "I don't usually get up late." "Neither do I." or "Me neither."
EXAMPLE: "If you don't do it yourself, no one will help you." "Neither would I."
5. Using 'Nor' without Neither
In addition to neither, we can also use nor in place of it.
EXAMPLE: "I never go to bed early." "Nor do I." or "Neither do I."
EXAMPLE: "I haven't seen the movie." "Nor have I." or "Neither have I."