1. May I? = Do I have permission?
We use "may" to ask permission to do something.
Examples
May I sit here? = Is it okay if I sit here?
May I speak to Mr Norman, please? Can I speak to Mr Norman, please? ("may" is more polite)
May she get a refund? Is she allowed to get a refund?
We can use "may" in the meaning of permission in positive sentences, too.
You may not speak loudly in a library. = The rules do not allow to do that.
2. It might rain. = It is possible that it will rain.
We use "might" (and "may", but not so often) to say that something is probable but we are not completely sure.
Examples
I might take a taxi or I might walk. I haven't decided yet. Both options are possible but I'm not sure what I'll choose.
Keep the phone with you. The delivery man might call any time. Maybe he'll call or maybe he won't call. We don't know.
Dan might not come tomorrow. He says he might have some other plans. It is possible that he won't come. Maybe he'll have some other plans. But he's not sure yet.