Function

Moral Obligation

We use should mostly to talk about what is the best thing or the most ideal thing to do in a situation:

The price of petrol should be reduced, it is too expensive.

You should finish your chores before you go out tonight

Sometimes, such moments can sound like a suggestion or even be worded as a question, a structure that is very common with parents talking to their children:

Do you think you should finish your homework first?

This is a form of providing the person an opportunity to "make the decision" for themselves or equally a question form has the direct obligation in plain view within the sentance : 

You should finish your homework

OR

Finish your homework!

Surprise or regret

We use shoudl have + participle to talk about things that were ideal but did not happen. As such it expresses regret:

I should have studied harder at university

We sometimes use Should to express surprise or regret about something. It is not a very common use and can sound a little old fashioned:

I am stunned that he should have done something so foolish

It is sad that it should end this way

Advice

We often use Should to give advice or make suggestions:

You should watch it at the cinema, it is totally worth it

We should start the unpacking tomorrow, it is late now

Predictions of things that are expected

We also use Should to predict what is likely to happen:

There should be a good crowd at the show tonight, the city has many fans of the band

He should be home any minute now, he siad the train was on time

Conditional Structures

Should is sometimes used in hypothetical (second conditionals) clauses with if to express possibility. It is a formal structure:

If you should wish to order breakfast, please inform the steward 

If you should decide to take the job, you must inform us within 28 days

To give thanks

In spoken English we often use you shouldn't have as a form of giving thanks when recieving a gift! It is a shortened version of the expression you shouldn't have done that/bought this

You got me a  pair of socks!  You shouldn't have!

Should and Ought to

Should and ought to have similar meaning and uses. Ought to is more formal and less common than should.

Should is far more common in negatives and questions

Should and Would

Should is a more fomal (and therefore less direct) alternative to would with I and we in conditional clauses