The "Either" as a adjective takes "singular countanle noun" i.e

Either road is long.

Either roads are long.(It is wrong.)

But,when we use either ... or like .....

Either the students or the teachers are reading the books.

Why "the students" ,which is plural form of "student" after Either.

Please help me.

In this case, "Either the students or the teachers are reading the books", either does not affect "students". The sentence could also read "Either the student or the teachers are reading the book". The first sentence means that many students or teachers could be reading the book. The second means that either one student or one of several teachers are reading the book.
As far as 'either' is concerned, you need to take two different approaches.
Approach 1: When you take 'either' alone, it is used as a pronoun, an adjective, or even an adverb.
Pronoun: (1)Either is fine.
(2)Either of the sisters is as tall as Joanne. (‘is’ is grammatically correct
though ‘are’ is often heard in reality; ‘sisters’ must be in the plural form
after ‘either of.’)
Adjective: (3)Either sister is as tall as Joanne. (‘sister’ must be in the singular form
when ‘either’ is used as an adjective.)
Adverb: (4)I don’t know, either. (‘either’ in this sentence is the same as ‘too’ in concept,
but the difference is: ‘either’ is used in the negative sentences, while ‘too’
in the affirmatives.)

Approach 2: When ‘either’ pairs with ‘or,’ the pair is categorized as a correlative conjunction.
In this case, you can enjoy the freedom of choosing either a singular or plural
noun after ‘either’ and ‘or.’ The choice depends solely on how many
things/people you are going to talk about. In addition, if this pair is a subject in
the affirmatives, the verb must be in agreement with the word appearing
after ‘or.’ But in the question sentences or inverted ones the verb must agree
with the word appearing immediately after ‘either.’ This is what they call the
principle of proximity.
(5)Either the teacher or his students are wrong.
(6)Either the students or their teacher is wrong.

Different opinions are welcome.