When do you use "might" or "may"?

Question

example : They may be going to the party

They might be going to the party

both sound pretty accurate to me. If both sentences ARE correct, do the meanings change somewhat because of the alternation between "may" and "might"?

Answer 1

I think generally the two words are interchangeable.  The difference between the two words is very subtle.  However, when asking for authorisation to do something, I was taught that it shows good manners to say 'Might I' rather than 'May I'.