Either means one of the two, while neither implies that none of the choices available are desirable. Either is generally applied as a determiner. It means one or the other as in” please complete the assignment by either Monday or Tuesday.” Another example is, “She is coming back home on Monday or Tuesday.”
Sometimes either is used as a determiner and in the sense of one or the other. Sometimes either is employed as a pronoun, alone or with a preposition. The word neither is often used before a singular noun when there is no article. For example, “Neither one of the candidates was a good option.” Another example is “Neither one of them knew who she was.” Neither is usually used before a singular verb, and it is often used in combination with “nor.”