Both, neither, either

We use both/neither/either when we talk about two things.

Imagine you're choosing which smartphone to buy and you're considering two options (A and B). You say:

"Both phones have good cameras." A has a good camera and B has a good camera, too.

"Neither phone is cheap." A is not cheap and B is not cheap.

"I can buy either phone online." I can buy A online or I can buy B online.

So, for short:

"both" = a AND b

"neither" = NOT a AND NOT b

"either" = a OR b

Both

"Both" is followed by a plural noun.

Both articles are informative.

We don't put "the" before "both", it's wrong.

I liked both films. WRONG the both films

We can drop the noun after "both" not to repeat it.

I couldn't decide between the black and the purple T-shirt, so I bought both.

We can say "both of" + the/these/my/Tom's/them. If "both" is followed by a noun (eg. these magazines, my parents, Nick's bikes), the preposition "of" becomes optional and we can drop it. If there is a pronoun, we put it into the objective case and we keep "of" (both of us, both of them). If the pronoun is the subject of the sentence, we can also say "we both", "they both".

Both of my friends travel a lot.

Both of Pat's daughters play the piano.

Both of those restaurants have Michelin stars.

I want to speak to both of you.

We both had a busy week.

There is no difference in meaning between both and both of. Compare:

Both employees were promoted.

Both of the employees were promoted.

Both of us knew what was going to happen.

We both knew what was going to happen.

If we want to name both things, we use "both ... and ..."

Both the president and the prime minister attended the press conference.

I was both thrilled and scared.

Neither and either

Neither/either is followed by a singular noun.

Neither task is urgent.

Neither student has answered correctly.

Either way is dangerous.

We can use neither/either alone not to repeat the noun.

What do you like more: broccoli or cauliflower? Neither.

Would you like green tea or black tea? Either. I don't care.

We can say neither/either of + the/these/my/Tom's/them.

Neither of these bags is heavy.

Neither of them wants to make a decision.

Neither of Lila's brothers lives in Canada.

I haven't met either of those men.

You can talk to either of us.

You can notice that neither contains negation in itself, so the verb is positive. But if we use a negative verb, we say "either". Compare with "no" and "any":

There are two green apples.