Just a few additional thoughts. Either refers to a choice of one option out of two available options; neither refers to a choice of zero options out of two available options. For example: "You can eat your cake either tomorrow, or the day after. Which will it be?" "Neither. I want it today". Or "Either. I don't mind". So you can use nor/or with the words, but you don't strictly "need" to.

It's how you use it in a sentence. "Neither" is negative and needs "nor". "Either" is positive and needs "or". So: Neither/nor: Not one, and also not the other Neither he nor I know English. Neither the dog nor the cat could enter the house. I did not know the answer, and neither did she. She did not want to draw, nor did she want to eat. Either/or: One or the other Either she or he needs to clean the house. Either you eat your dinner, or you will get no dessert. She wanted either the purple or the red dress. He could write his exam, or he could play video games.

Neither = not + either.