When offered a choice between two things you don't like, you might choose neither. Whereas "either" means you would pick one, neither closes the door on both, saying you do "not" choose "either" one.
The adjective neither comes directly from Middle English, evolving from the word nawther. Neither is versatile, and can be used as a pronoun, ("Neither of your ideas works"), or an adjective ("Neither plan can be used"). It can also be used as a conjunction, in which case it is usually paired with "nor" ("We can use neither your first suggestion nor your second one"). Sounds like you're having a tough day.
