Either... or is used for ONE thing, but NOT the other.
You can choose one flavor of ice cream - either chocolate or vanilla.
We can either go shopping or see a movie, but we won't have time to do both.
Neither... nor is used for NOT TWO THINGS. You can also say Neither of + two things:
I don't like soccer. I don't like tennis. I like neither soccer nor tennis.
I have two brothers. Neither of my brothers have blue eyes.
Neither is also used for agreeing with a negative comment. A "negative comment" is any phrase with a negative auxiliary verb: don't, didn't, haven't, can't, won't, etc.
"I like strawberry ice cream." (positive comment)
"Me too!"
"I don't like strawberry ice cream." (negative comment)
"Me neither."
"I've been to Europe several times." (positive comment)
"Me too."
"I've never been to Europe." (negative comment)
"Me neither."
"I can speak Chinese." (positive comment)
"Me too."
"I can't speak Chinese." (negative comment)
"Me neither."
The phrase "Me either" is not technically correct, but many people say it in spoken English instead of "Me neither"!