may 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
might 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.