Pronouns used as complements of to be

Grammarians formerly recommended that a pronoun used as the complement of the verb to be should be in the nominative case. Today the use of the nominative case in such cases is considered extremely formal and over-correct. Instead, we use the objective case.

It is me. (Formal: It is I.)
It was him. (Formal: It was he.)

A pronoun used as the object of a verb or a preposition should be in the objective case.

You can’t trust him. (NOT You can’t trust he.)
We have invited them. (NOT We have invited they.)

There is really no difference between you and him. (Here the pronouns you and him are used as the objects of the preposition between.)

He has given great trouble to us. (Here we use the objective case because the pronoun us is the object of the verb to.)

A pronoun must agree with its antecedent in number, person and gender.

All students should bring their books. (Here the pronoun their agrees with its antecedent students in number and person.)
John has brought his book. (Here the pronoun his agrees with its antecedent John in number, person and gender.)
Each of the girls gave her own version of the story.
I am not one of those who believe everything they hear. (Here the antecedent of the pronoun they is those and not I.)

Some grammarians recommend that the pronoun of the masculine gender should be used to refer back to anybody, everybody, anyone, each etc., when the sex is unknown.

Everybody ran as fast as he could.
Anybody can do it if he tries.

In modern English it is more common to use plural pronouns to refer back to anybody, everyone etc.

Everybody should bring their books. (Less formal)
Everybody should bring his books. (Very formal)
Everybody ran as fast as they could.
Each of them had their share. (Less formal)
Each of them had his share. (Very formal)

Who and whom

Who is in the nominative case; whom is in the objective case.

I don’t know who (not whom) they were.
The student, whom you thought so highly of, has failed to win the first prize.

In modern English whom is unusual except in a formal style.

Who did you meet? (Less formal)
Whom did you meet? (Very formal)