Paired conjunctions (or correlative conjunctions) – both…and, either…or, neither…nor, not only…but also – join 2 things.  Each of these four has its own usage.

Both … and:

The first of the paired conjunctions – both … and – connects 2 positive things, such as noun subjects, noun objects, verbs, adjectives, adverbs, prepositional phrases, and other words.  You must only connect things that are the same.  The first part of the pair (both) can be dropped with no change in meaning.

Examples:

Both her mother and her sisters want her to marry a rich man.
Her mother and her sisters want her to marry a rich man.

I like both apples and oranges.
I like apples and oranges.

She both skis and snowboards.
She skis and snowboards.

The man is both kind and generous.
The man is kind and generous.

The subway moves both fast and quietly.
The subway moves fast and quietly.

That color is both in the dining room and in the living room.
That color is in the dining room and in the living room.

Either … or:

The second of the paired conjunctions is either … or.  If it is used in a positive sentence, then the things that follow are choices and only one of them can be true.  It is used with noun subjects and noun objects, verbs, adjectives, adverbs, and prepositional phrases, and other words.  Either can be dropped with no change in meaning

Examples:

Either the principal or the teachers are giving the students a day off.  (Only one of them is giving the students a day off.)
The principal or the teachers are giving the students a day off.

NOTE:  The subject closest to the verb determines if the verb is singular or plural.
Either the teachers or the principal is giving the students a day off.

I like either apples or oranges for an afternoon snack.  (I like both but I only choose one.)
I like apples or oranges for an afternoon snack.

She either skis or snowboards on the weekends.  (She does both but only chooses one at a time.)
She skis or snowboards on the weekends.

The man is either brave or stupid; I don’t know exactly which one.  (The man could be both but is only one of those.)
The man is brave or stupid; I don’t know exactly which one.

The subway moves either fast or slowly depending on the volume of passengers.  (It does only one of those at a time – fast or slowly.)
The subway moves fast or slowly depending on the volume of passengers.

My briefcase is either in the dining room or in the living room.  (It is only in one of those places.)
My briefcase is in the dining room or in the living room.

If either … or is used in a negative sentence, then both things that follow are true.  It is only used with noun objects or adverbs.  Either can be dropped with no change in meaning.

Examples:

I don’t eat either bread or pasta.  (I don’t like bread; I don’t like pasta.)
I don’t like bread or pasta.

The subway doesn’t move either fast or quietly.  (It doesn’t move fast; it doesn’t move quietly.)
The subway doesn’t move fast or quietly.

NOTE:  With verbs, adjectives, and prepositional phrases either is always dropped:

Examples:

I don’t take a nap or meditate in the late afternoon.
NOT: I don’t either take a nap or meditate in the afternoon.

The man isn’t kind or generous.
NOT: The man isn’t either kind or generous.

My briefcase isn’t in the dining room or in the living room.
NOT: My briefcase isn’t either in the dining room or in the living room.

NOTE:  The easiest way to learn this grammar is to not use either … or in negative sentences but only in positive ones.

Neither … nor:

The third of the paired conjunctions is neither … nor.  Although the meaning is “not this and not that,” it is always used in positive sentences, never in negative ones.  It is used with noun subject, noun objects, verbs, adjectives, adverbs, prepositional phrases, and other words.

Examples:

Neither the principal nor the teachers are supervising the students.

NOTE:  The subject closest to the verb determines if the verb is singular or plural.
Neither the teachers nor the principal is supervising the students.

I like neither apples nor oranges.
SAME AS:  I don’t like either apples or oranges.

She neither skis nor snowboards.
SAME AS:  She doesn’t ski or snowboard.

The man is neither kind nor generous.
SAME AS:  The man isn’t kind or generous.

The subway moves neither fast nor quietly.
SAME AS:  The subway doesn’t move fast or quietly.

That color is neither in the dining room nor in the living room.
SAME AS:  That color isn’t in the dining room or in the living room.

Not only … but also:

The fourth of the paired conjunctions – not only … but also – connects 2 things, such as noun subjects, noun objects, verbs, adjectives, adverbs, prepositional phrases, and other words.  The second part of the pair (but also) can be separated with no change in meaning.  When but is separated from also, also can change to too.

Examples:

Not only did he fail math, but he also failed English.
Not only did he fail math, but he failed English also.
Not only did he fail math, but he failed English too.

NOTE:  When not only starts a sentence, the subject and verb are inverted (switched around.)
SAME AS:  He failed (both) math and English.

We like not only Japanese food but also Korean food.
We like not only Japanese food but Korean food also.
We like not only Japanese food but Korean food too.
SAME AS:  We like (both) Japanese food and Korean food.

They not only travel a lot, but they also make travel videos.
They not only travel a lot, but they make travel videos also.
They not only travel a lot, but they make travel videos too.
SAME AS:  They (both) travel a lot and make travel videos.

People consider him not only intelligent but also very creative.
People consider him not only intelligent but very creative also.
People consider him not only intelligent but very creative too.
SAME AS:  People consider him (both) intelligent and very creative.

She plays poker not only loosely but also aggressively.
She plays poker not only loosely but aggressively also.
She plays poker not only loosely but aggressively too.
SAME AS:  She plays poker (both) loosely and aggressively.

Mike made breakfast not only for himself but also for his wife.
Mike made breakfast not only for himself but for his wife also.
Mike made breakfast not only for himself but for his wife too.
SAME AS:  Mike made breakfast (both) for himself and his wife.

NOTE:  Some people like to put a comma ( , ) before too.
Mike made breakfast not only for himself but for his wife, too.