That age-old dispute about the pronunciation of the words 'either' and 'neither' as battled out in the famous song would perhaps be best left alone given that, at the end of the day, both are correct and it is merely a matter of regional accent. However, it is not always so easy to determine the usage of these two words together with their trusty friends or and nor...

A first pointer to retain is that the phrase 'either of' always refers to one or other of two things and should therefore never be used with more than two. For example, you should say (or write):
"I haven't read either of the two reports"
but
"I haven't read any of the three reports".
Also, when using 'either of' followed by a verb, again as it refers to one of two things, the verb should always be in the singular, e.g.:
"Either of the two reports is suitable."
These same rules apply when writing (or saying!) 'neither of'.

As to the pairs, let us begin with the more positive of the two: either and or. The rules for usage here are relatively simple. This pair is used to determine a choice between two things, such as in, "you can take your holidays either for one week in July or for two weeks in June". Dutch and French speakers alike please note that you cannot use or followed by or, even if the translation is tempting directly from 'ofwel ... ofwel' or 'soit ... soit'.

Things become much more complicated, however, as soon as negatives come into play. The most common pair is neither and nor, indicating that no one of two things is applicable, such as in, "her telephone manner was neither polite nor co-operative". However, nor can also be used after no, as can or (!). This means that both "she had no manners nor patience" and "she had no manners or patience" are correct. Equally, both can be used after not, as in the example, "he has not yet written the minutes (n)or the agenda". A tendency towards or would be preferred given that the negativity of no and not carries forward anyway. It should also be noted that you can feel free to use nor more than once after neither if you wish to indicate that no one of several things is applicable, e.g. "this desk is neither tidy nor organised nor is it properly equipped!" (note the inversion of the verb following nor here).

There are other, often more oral uses of these two words. Neither is used to indicate, for example, that you did not do something that some one else has already stated they did not do:
"I didn't go to the cinema yesterday."
"Neither did I."
But note that either can be used here with not to the same effect:
"I didnХt either."
This is a construction that is often clumsy, but difficult to avoid, in written English. A handy hint is to always remain positive and say instead, for example:
"It was confirmed that the second project was also not a good idea."