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

UseExamplesExamples
possibilityIt may rain today.
permission to do sth. in the presentMay I sit here?Am I allowed to sit here?
polite offerMay I help you?

2. might

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