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1) Both
Both means two of two things.
I have two cats. I like both of them.

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2) Neither
Neither means not one or the other of two things.
Neither of my cats is grey.
Remember to use a singular verb after neither.
Neither of the dogs are dangerous. => Neither of the dogs is dangerous.

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3) Either
Either means one or the other.
There are two cakes. Please have one. You can have either one.

Form:

1) You can use both, neither and either directly before a noun.
Both supermarkets are good.
Neither supermarket sells electrical goods.
We can go to either supermarket, I don’t mind.

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2) Both, neither and either are often used with ‘of’. But you must always use a determiner (the, my, these, those, his etc) before the noun.

Both of children like chocolate cake. => Both of the children like chocolate cake.

However, you don’t have to use of with both.
Both of the children like chocolate cake. 
Both children like chocolate cake.

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3) You can use both, neither and either+ of + object pronoun(you, them, us).

Both of them wore white dresses.  
Neither of us was late.  
Have either of you got a pen?

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4) You can use both ... and ...; neither ... nor ..., and either ... or ....
Examples:
Both James and Diana work here.
Neither James nor Diana works here.
You can ask either James or Diana