We can use these words in noun phrases. We use them when we are talking about exactly two (not one and not more than two). Both is plural (Both girls are married); either and neither are singular (Neither of them has children).
There were cars parked on both sides of the road.
I have two sons. Both of them are engineers.
There was no space to park on either side of the road.
I've got two cars, but I don't really like either of them.
Neither side of the street was lit. It was very dark.
Neither of my two sons is married yet.
Both Argentina and West Germany have won the World Cup twice.
(Argentina has won the World Cup twice and West Germany has won it twice.)
Either Argentina or Brazil will win the next World Cup.
(Argentina might win or Brazil might win; I don't think another team will win.)
Neither Scotland nor Wales has ever won the World Cup.
(Scotland has never won it and Wales has never won it.)
NOTICE: In speaking only, we often use a plural verb with neither.
Neither of my sons are married yet.
Neither Scotland nor Wales have ever won it.