Might

Possibility

We use "Might" to express a slight or weak possibility. 

Perhaps we shouldn't eat at the curry house, your guests might not like spicy food

We might be able to afford a trip to the beach if I get paid in time. 

We use "Might" + have + participle to express such possibilities that did not come to pass in the past:

We might have been able to afford a holiday but I didn't get paid.

Permission

"Might" can be used in a very formal way to make enquiries or ask permission. It is much more common to use can, could or may.

Might I ask where you bought that shirt?

Might I enquire as to his whereabouts?

Suggestions

We can use "Might" to give advice or make suggestions sound more polite, especially when used with verbs such as "like", "prefer" or "want":

You might like to reconsider your course of action

He might want to try our home delivery service

Criticism

We use "Might" + have + participle to express criticism or disapproval of an action

You might have told us you were not coming for Christmas

He might have tried to explain how he felt rather than simply leaving

Reporting "May"

When we report speech, some verbs are changed. "May" is reported as "Might"

"She may have had a change of heart" siad Brian.

Reported as:

Brian said she might have had a change of heart,