We can use 'may' to ask for permission. However this is rather formal and not used very often in modern spoken English.
May I leave now?
May I borrow your dictionary?
May we think about it until tomorrow?
We use 'may' to suggest something is possible.
Philip may come to stay with us
I may not have time to do it straightaway.
It may snow later today.
We use 'might' to suggest a small possibility of something. Often we read that 'might' suggests a smaller possibility that 'may', there is in fact little difference and 'might is more usual than 'may' in spoken English.
It might rain this afternoon.
She might be at home by now but it's not sure at all.
I might not have time to go to the shops for you.
I might not go.
For the past, we use 'might have'.
He might have tried to call you while you were out.