The verbs like can, could, may, might, would, shall, should and ought are called modal verbs or modals. They are used with ordinary verbs to express meanings such as possibility, permission, certainly, etc.
(1) Can usually expresses ability or capacity
I can swim across the river.
Can you lift this table?
(2) Can is also used to express permission
You can go now.
(3) May is a more formal modal used to express permission
You may come in.
May I leave the room now?
(4) May is also used to suggest possibility in an affirmative sentence.
He may be at home
It may rain tomorrow
(5) Can is used to suggest possibility in negative/interrogative sentence.
Can this be true?
It cannot be so.
(6) May when used in a negative sentence suggests an improbability whereas can suggests impossibility.
He may not come today.
She cannot sing.
(7) Could and might are used as past tense forms of can and ‘may’.
I could swim across the river when I was young.
I thought he might be at home.
(8) Might suggests less possibility or probability than may.
I might go to Bangalore next week suggests the probability of going is less than a sentence with ‘may’ will suggest.
(9) Could is used as a polite form of seeking permission or making a request.
Could you pass me the plate?
Could I please talk to Mr. Grover?
(10) Shall is used with first person and will in all the persons to denote future action.
I shall need the money tomorrow.
When will you come next?
(11) Shall is used with the second and third person to express command, promise or threat.
You shall never come near my child.
You shall be punished for this.
We shall go for a picnic this Sunday.
(12) Will You? Indicates an invitation or request.
Will you dine with us tonight?
Will you lend me your car for a week?
(13) Should and would are used as past forms of shall and will.
I expected that I would get a first class.
She would sit for hours listening to the radio.
(14) Should is used to express duty or obligation.
We should obey the laws.
You should keep your promise.
(15) Should is used to express a supposition
If it should rain, they will not come.
(16) Should can also be used to express probability.
He should be in the library.
(17) Must is used to express necessity.
You must improve your spelling.
(18) Must is also need to express obligation, and is a stronger word than should
We must follow the law.
(19) Must is also used to express logical certainty.
Living alone in such a big city must be difficult.
(20) Ought is used to express moral obligation and is stronger than both should and must.
We ought to love our parents.
(21) Ought is also used to express probability sometimes when the probability is very strong. The book ought to be very useful.