We can use 'may' to ask for permission. However this is rather formal and not used very often in modern spoken English
- May I borrow your pen?
- May we think about it?
- May I go now?
We use 'may' to suggest something is possible
- It may rain later today.
- I may not have time to do it today.
- Pete may come with us
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.
- She might be at home by now but it's not sure at all.
- It might rain this afternoon.
- 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 while I was out.
- I might have dropped it in the street.