Each set refers to a choice between two items [A and B] independently.
Either A or B is true. A is true or B is true, but both are not true.
Neither A nor B is true. Both are false.
"Nor" is the word that is most commonly misused. The following is a usage note from the AHD:
USAGE NOTE When using neither in a balanced construction that negates two parts of a sentence, nor (not or) must be used in the second clause: She is neither able nor (not or) willing to go. Similarly, when negating the second of two negative independent clauses, nor (not or) must be used: He cannot find anyone now, nor does he expect to find anyone in the future; Jane will never compromise with Bill, nor will Bill compromise with Jane. Note that in these constructions, nor causes an inversion of the auxiliary verb and the subject (does he … will Bill …). However, when a verb is negated by not or never, and is followed by a verb phrase that is also to be negated (but not an entire clause), either or or nor can be used: He will not permit the change, or (or nor) even consider it. In noun phrases of the type no this or that, or is actually more common than nor: He has no experience or interest (less frequently nor interest) in chemistry. Or is also more common than nor when such a noun phrase, adjective phrase, or adverb phrase is introduced by not: He is not a philosopher or a statesman. They were not rich or happy. See Usage Notes at neither, or1.