To express permission

May is used to ask and give permission.

May I come in, Sir?
Yes, you may.
May I go now?
No, you may not.

Note that can is also used to express permission. And now-a-days the denial of permission is often expressed by cannot, and not may not.

May we play tennis, Mummy?
No, you can’t. (More common than No, you mayn’t.)

This usage is encouraged by the fact that the contraction can’t is easier to say than mayn’t.

To express possibility

May can be used to express possibility.

She may or may not agree with me.
He may come.
She may be elected President of the club.

Notes:

Can is also used to express possibility, but the usage is different. While can expresses theoretical possibility, may is used to indicate factual possibility.

Compare:
Accidents can happen any time. (Theoretical possibility: It is possible for accidents to happen any time.)
He may soon have an accident if he continues to drive carelessly. (Factual possibility: He doesn’t drive carefully, so it is factually possible for him to have an accident.)
This road can be blocked. (It is possible to block this road. In fact, any road can be blocked.)
This road may be blocked today due to the procession. (Factual possibility)

To express a wish

May is used in expressing a wish.

May his soul rest in peace!
May God bless you!

To express purpose

May is used in subordinate clauses that express purpose.

Use fertilizers so that you may have a rich harvest.
Work hard that you may pass.
Eat that you may live.

Might

Indirect speech

Might is the past tense of may in indirect speech.

He said, “I may come.”
He said that he might come.
The boy said, “I may have done so.”
The boy said that he might have done so.

To express possibility

Might is often used to express possibility. Note that might expresses less possibility than may.

Compare:
I may pass. (50% chance)
I might pass. (20% chance)
The patient may recover. (It is possible for the patient to recover.)
The patient might recover. (It is possible but very unlikely.)