Correlative conjunctions

Both … and
Subjects connected by both … and take the a plural conjugation.
Both Sharon and her children live in Fresno.

Either … or
"one or the other, this or that, he or she, etc." Verb conjugation depends on the subject (singular or plural) closest to the conjugated verb.
Either Peter or the girls need to attend the course. (second subject 'the girls' plural)
Either Jane or Matt is going to visit next weekend. (second subject 'Matt' singular)

Neither … nor

Neither … nor is used in sentences in a negative sense meaning "not this one nor the other, not this nor that, not he nor she, etc.".

Verb conjugation depends on the subject (singular or plural) closest to the conjugated verb.

Neither Frank nor Lilly lives in Eugene. (second subject 'Lilly' singular)
Neither Axel nor my other friends care about their future. (second subject 'other friends' plural)

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The forms both ... and, and either ... or can also be used as objects of verbs . In this case, there is no need to conjugate verbs.

I'm going to have both steak and eggs for breakfast.
They moved to either Seattle or Chicago. I can't remember which.
I enjoy both golf and tennis.