Either vs Neither ( English )

As adjective:
‘Either’ indicates one or the other, or both. 
For example:
You may use either hand for the purpose. = You may use your right or left hand for the purpose.
There were tall houses on either side of the river. = There were tall houses on both sides of the river.
‘Neither’ indicates not one or the other; none of the two.
For example:
Neither twin was invited to the wedding. = None of the twins was invited to the wedding.

As pronoun:
‘Either’ indicates one or the other.
For example:
Both buses are headed in that direction, you can get on either. = Both buses are headed in that direction, you can get on one or the other.
‘Neither’ indicates not one or the other.
For example:
Both pups were pure-breed, but neither displayed the characteristic traits of its breed. = Both pups were pure-breed, but not one or the other displayed the characteristic traits of its breed.