Modal verbs: May - Might

May and might are used to say that something is a possibility. 

May
The modal verb may is manly used in written and formal contests, is rather formal and not used very often in modern spoken English.

Is possible to use it:

• to ask for a permission:  

      May I have a cup of tea?  
        (this is the typical English contest we could imagine, like at 5 o'clock in the hall of a Lord's club)  
      May I borrow your pen? (in an office)

• to suggest something is possible:  

      I may not have time to do it today. (in an office)

Might
Might suggests a smaller possibility than may, there is basically just a little difference and is more usual than 'may' in spoken English.

• the modal verb might is used when the outcome is less probable or uncertain to happen:  

        Tom is not in the office, he might not work here anymore.
       (he is not in the office, but maybe he is just having a day off)


more examples, in past version as well:
- I can't find my wallet anywhere. I might have left it in the car.
- I might not go to the party.
- I wonder why Dave didn't answer the phone, he might have been sleeping.


MORE:
Could is similar to may and might:
- What you've just said is a weird story, but it could be true = (it may/might be true)
- You could have forgotten your wallet at home = (you may/might have forgotten)

BUT remember that couldn't is different from may not and might not, compare:
- Police said it couldn't have been an accident. (is propable that was something planned)

- He didn't answer me. I might not have been English. (is probable that he wasn't able to understand me)


more examples:
- I haven't decided where to go next weekend, i might go to Milan. 
  (perhaps I will go to Milan = future plans)

- I might not go out tomorrow evening, I don't feel good. 
  (perhaps I will not go out = future plans)

- Don't phone at my parents' house at 11 o'clock, they might be already sleeping = Don't phone at my parents' house at 11 o'clock, they will be already sleeping