May and might are both used to talk about the chances of something happening and to ask for and give permission. They are mainly used to talk about the present and future. The difference between them is that might is used for less probable things and may for more probable:

I may go to London tomorrow (50% chance)

I might go to London tomorrow (30% chance)

  • Requests

Both are very formal and Might is more polite than May

Might/May I have a glass of water? 

  • Permission

As May is stronger than Might permission is given with May.

Might/May I have a glass of water? Yes, you may. (not Yes, you might)

If we want to talk about permission in the past we use could/be allowed to.

1. may

Use Examples Examples
possibility It may rain today.
permission to do sth. in the present May I sit here? Am I allowed to sit here?
polite offer May I help you?

2. might

Use Examples
possibility (less possible than may) It might rain today.
permission to do sth. in the present Might I sit here?
very polite offer  Might I help you?