Rule 1

A personal pronoun must agree with its antecedent (the noun it refers back to) in number, person and gender.

John is a good student. He stood first in the examination.
Alice is my sister. She lives abroad.
The children had been standing in the sun. They were thirsty.

Here the pronouns he, she and they are of the same number, person and gender as the nouns John, Alice and children respectively.

Rule 2

When two singular nouns connected by and refer to the same person, the pronoun should be singular. But when they represent different persons the pronoun should be in the plural.

My father and guardian has given his consent to my plan. (Here the nouns father and guardian refer to the same person.)
The Secretary and the treasurer were asked to settle their differences quickly. (Here the nouns Secretary and Treasurer refer to different persons.)

Rule 3

When two singular nouns connected by and are preceded by each and every, the pronoun must be in the singular.

Each Admiral and Each General did his duty well.

Rule 4

When two singular nouns are joined by or, either…or, neither…nor, the following pronoun should be in the singular.

Neither John nor Peter did his homework.
Either Alice or Mary will bring her guitar.

Rule 5

When a singular noun and a plural noun are joined by or or nor, the pronoun should agree with the nearest noun.

Neither the officer nor his subordinates did their duty well.

Rule 6

The pronoun following than must be in the same case as the noun or pronoun preceding it.

You are taller than he. (NOT …him.)
He ran faster than I. (NOT …me.)

Note: This rule is no longer strictly followed. Many educated people now use objective pronouns after than.