1.
Meanings associated with modals
be able to
to show ability: I am able to write English well.
can
to show ability: I can write English well.
to give permission: You can leave when you finish work.
to make requests: I asked my friend if she can help me revise for the examination.
could
to show ability: I could write English well when I was in secondary school, but now I am
out of practice.
to show possibility: It could rain today.
to give conditional permission: You could leave if you had finished your work.
to make requests: I asked my friend if she could help me revise for the examination.
had better
to offer advice: You had better work harder.
may
to show possibility: It may rain today.
to indicate probability: This may be the correct answer.
to give permission: You may leave when you finish the work.
might
to show possibility: It might rain today.
need
to ask if there is a reason to do something or explain that there is no reason: Need I
come? ... No, you need not come.

2.
ought to
to offer advice: You ought to work harder.
shall
to indicate intention, make promises or predication: I shall return. (absolute promise or
certainty)
should
to offer advice: You should work harder.
used to
to show familiarity: I got used to his humour.
will
to make requests: I'll ask my friend if she will help me revise for the examination.
to indicate intention, make promises or predications: I will pass easily this test.
why don't
to make suggestions: why don't we go swimming.
would
to make requests: I asked my friend if she would help me revise for the examinations.
to show preference or desire: I'd like to miss this class. / I'd rather not have another
piece of cake.
to present an offer: Would you like to meet at the coffee shop?
You can learn more about modals by going to HKUST's English Grammar Guide.