Either and neither have two different means, and be used in different ways.
1) As an adjective
Either means one or the other of the two things.
Neither means not one or the other two of things.
e.g.: - I would buy either bags (= I'd buy one or the other of the two bags)
- I would buy neither bags (= I wouldn't buy either of the bags)
2) As an pronoun (you need to add the prepostion 'of' )
Either means the one or the other.
Neither means none of them.
e.g.: - I want to buy either of the books (= I want to buy the one book or the other book)
- I want to buy neither of the books (= I don't want to buy either of the books)
3) As a conjunction
Either... or... used to indicate possibility or choices.
Neither... nor... used to indicate something is not true.
e.g.: - I'll go to either Japan or Korea (= I'll choose one of them)
- I'll go to neither Japan nor Korea (= I won't go to either of these two countries)
4) As an adverb
Either has the same meaning with 'too', which is 'in addition'. Either can only be used in a negative statement.
e.g.: - I don't like dogs, I don't like cats either.
- The apartment is big, it's not expensive either.
However, either and neither has the same meaning when are used in informal speech.
e.g.: - She can't eat meat, me either (= I can't eat meat)
- He didn't come last night, me neither (I didn't come last night)