Should and ought have very similar meanings. In fact, these words can often replace each other.

She should be more responsible. OR She ought to be more responsible.

Both should and ought are used to talk about obligation and duty, to give advice and to say what we think it is appropriate for people to do. Note that should is much more frequent than ought.

You should cultivate the habit of reading. OR You ought to cultivate the habit of reading.
You ought to / should have been more careful.

Note that should and ought are not used in polite requests. Instead we use a question form with could.

Could you move a bit? (NOT You should / ought to move a bit.)

Both should and ought can be used to talk about logical possibility.

I have bought ten eggs – that should be enough.

Note that should is followed by an infinitive without to. Ought is followed by an infinitive with to.

You should wait. (NOT You should to wait.)
You ought to wait. (NOT You ought wait.)

Must is stronger than should and ought. Must is used when the speaker is confident that something is true or something will happen.

Compare:

You must stop smoking. (An order which is likely to be obeyed.)
You ought to stop smoking. (A piece of advice that may or may not be followed.)