May Vs Might

'Might' is used mostly to express possibility. English speakers use 'might' to make suggestions or requests, although this is more common in British English and could be seen as extremely formal.
'Might' is also used in conditional sentences.

Possibility: Your phone might be in the kitchen.
Suggestion: You might want to suggest that to the committee.
Request: Might I ask a question?
Conditional: If I didn't have to study, I might go out.

'Might' like most modals is very different in the past or future. Here are some examples of 'might' with its different functions:

Possibility

Present: Peter might be stuck in traffic. / Peter might not be stuck in traffic.
Past: Peter might have been stuck in traffic. / Peter might not have been stuck in traffic.
Future: Peter might be stuck in traffic when he leaves work. / Peter might not be stuck in traffic when he leaves work.

With possibility 'could' and 'may' are also possible.

Suggestion

Past: You might have caught a taxi.
Future: You might want to catch a taxi. / You might not want to catch a taxi.

With suggestions 'could' is also possible. With suggestions using 'might' there is no present form and no negative form in the past.

Request

(more common in British English)

Present: 
Might I have a coffee?
Might I ask who has been invited to the party?

Requests usually refer to an immediate future. 'could', 'may' and 'can' are also possible.

May

'May' is mostly used to express possibility. It can also be used to ask and to give permission although this function is becoming less common.

Alex may be at home. Possibility
You may leave once you’ve finished the assignment. Giving permission
May I use your pen? Requesting permission.

For possibility 'may' has the same meaning as 'might' or 'can'.
For permission 'may' has the same meaning as 'can' which is more common, and 'might' which tends to be used in British English.