In Indirect Speech

Would is the past tense form of will in indirect speech.

Direct Speech: The teacher said, “The school will be closed on Monday.”
Indirect Speech: The teacher said that the school would be closed on Monday.
Direct speech: She said, “I will help you.”
Indirect speech: She said that she would help me.
Direct speech: He said, “I will look into the matter.”
Indirect speech: He said that he would look into the matter.

To express willingness and determination

Would can be used to express willingness and determination.

She said that she would accept the offer. (Willingness)
They said that they would do everything to help me out. (Willingness)
They would cut classes in spite of the repeated warnings given by the principal. (Determination)

To talk about a habitual or customary action in the past

Would can be used to talk about a habitual or customary action in the past.

After dinner they would sit in the hall and chat for quite sometime.
The old woman would spend most of the time talking to herself.
He would go for a long walk every morning.

Would and Would like to

Both would and would like to are used to express wishes. Note would like to is more polite than would.

I would know what I am supposed to do.
I would like to know what I am supposed to do.
Behave towards others as you would like them to behave towards you.
I would like to ask you something.

In polite questions

Would is often used for asking polite questions:
Would you like to go for a walk? (More polite than Will you like to go for a walk.)
Would you mind my smoking? (More polite than Will you mind my smoking.)
Would you, please, hire me a taxi?
Would you mind lending me your bicycle for a few hours?
Would you, please, close the door?

With if clauses

Would can be used in result clauses preceded or followed by a conditional clause expressing an impossible or improbable condition.

If I could fly, I would be with you now.
Were I the Prime Minister, I would try to eradicate poverty and illiteracy.

Would rather

Would rather is used to express choice or preference.

I would rather starve than beg.
I would rather go to jail than pay the fine.
She would rather die than marry him.