"Neither" is the negative equivalent of "either", in exactly the same way that 'nothing' is the negative equivalent of 'anything', and 'nobody' is the negative equivalent of 'anybody', and so on.
Either = A or B
"Do you want apple juice or orange juice?"
"Either is fine. I don't mind."
Neither = Not A and not B
"Do you want want pizza or spaghetti?"
Neither. Sorry, but I'm allergic to wheat.
"Do you want apple juice or orange juice?"
Either is fine. I don't mind.