The difference between would and should

The verbs should and would are modal auxiliaries. There are some differences between them.

In general would and should are used as the past form or less definite form of will and shall.

Will you come in?
She asked me if I would come in.

I told him that I should see him the following day.

Should

Should is used to talk about obligation.

I should go now.
He should go now.
They should go now.
You should go now.

Would

Would is used to talk about past habits.

The old woman would keep talking about her illnesses.
When we were kids we would spend hours playing cricket.

Should / would

Would is used with first, second and third person pronouns in conditional statements and questions.

If a car had no brakes there would soon be an accident.
I told them that we would probably be late.
If you would come this way, we could go for a drive.

Should

Should is used with first person pronouns in conditional sentences.

I should be grateful for an early reply. OR I would be grateful for an early reply.
We should probably come. OR We would probably come.

Would and should are also used in requests, questions and offers.

Would you like some coffee?
Would you mind my smoking?