Rule 1

The verb must agree with its subject in number and person. In other words a singular subject should be followed by a singular verb. Similarly, a plural subject should be followed by a plural verb.

I am going to the library.
She is writing letters.
We are having dinner.
It is raining.
You are a nice chap.
It was a cloudy day.
John is my brother.
John and Susie are close friends.

Notes:

In the present tense, we use am with I, is with singular nouns/pronouns (e.g. he, she, it, Peter, Alice) and are with plural nouns/pronouns (e.g. they, we, you, children, trees, birds etc.). In the past tense, we use was with I and other singular subjects and were with plural subjects.

The pronoun you can be singular or plural in number, but it is always followed by the plural verb are/were.

Rule 2

The pronouns another, anything, each, everyone, everybody, anybody, someone, somebody, no one, either and neither are singular. Hence, they should be followed by singular verbs.

Everyone in the class has handed in work. (NOT Everyone in the class have …)
Neither of the girls seems to be correct. (NOT Neither of the girls seem …)
Each of the boys was given a watch. (NOT Each of the boys were given …)

Rule 3

Group words or collective nouns take a singular verb if you are talking of the group as a whole. They take a plural verb if you are talking about the individuals within the group.

His family lives in this house. (Here the collective noun family refers to the group as a whole.)
His family live in various parts of Paris. (Here we are talking about the individual members of the family.)
The jury is in the courtroom. (Singular verb because we are referring to the group as a whole)
The jury are still debating the case. (Plural verb because we are talking about the individual members)

Note that collective nouns are always followed by singular verbs in American English.

Rule 4

Subjects comprising two or more nouns/pronouns will take a singular verb if the nouns together express one idea.

Slow and steady wins the race.
Fish and chips is popular in England.
Bacon and eggs was served for breakfast.

Rule 5

When a relative pronoun is the subject of the clause, it must agree in number and person with its antecedent (i.e. the noun it stands for). Note that the verb, in this case, must agree with the antecedent of the relative pronoun.

She is one of the best girls who have joined this institution. (Here the verb have agrees with the antecedent girls.)
I, who am already late, cannot afford to stay any longer.
You, who are my friend, must listen to what I have to say.

Rule 6

When a group of words containing a plural noun represents a single subject, you must use a singular verb.

The Adventures of Tom Sawyer is my favorite book.