This is a bit of a tricky one, as English speakers don’t always agree on the difference between “may” and “might”. For some, there is a small difference between the two, and for others, there is no difference at all. I’ll stick to the dictionary definitions and try to provide some examples which are simpler than the ones you find if you Google this question. I’ll give you some advice in bold.
To express likelihood
Let’s have a look at what the Cambridge Dictionary says:
The restaurant may close
It is likely that the restaurant will close
The speaker isn’t sure but thinks there is a strong chance that the restaurant will close.
The restaurant might close
There is only a possibility that the restaurant will close but no one is very sure
The speaker isn’t sure but thinks there is a chance the restaurant will close.
So here, the difference between ‘may’ and ‘might’ is likelihood (how likely something is to happen), with ‘may’ being the stronger of the two. Many speakers don’t see much difference though, so don’t worry about getting this wrong!
‘May’ and ‘might’ are commonly used to express uncertainty in the past:
“She may have dropped her keys in the road”.
“She might have dropped her keys in the road”.
I don’t see that there’s much difference between the two here, though it’s more common to use ‘might’ in the past tense and ‘may’ in the present tense. Remember that in informal written English and spoken English these would be contracted to may’ve /meɪv/ and must’ve /ˈmʌstəv/.
To ask permission
I’m sure you’ve seen or heard questions using ‘may’, like this:
A “May I use your bathroom?”
B “Of course (you may)! It’s down the hall, first door on the right”.
It’s possible to use ‘might’ instead of ‘may’, but it’s rather formal and not very common these days. Stick to using ‘may I..’ because this is correct, polite, and used frequently.
Just be a little careful with following sentences:
“Alice may not go to the party” – Alice either doesn’t have permission to go to the party or there is a chance she will not attend the party.
“Alice might not go to the party” – There is a chance that Alice will not attend the party.
To give advice
Rather than saying ‘you should’ to give advice, it’s more often more polite to use ‘might’:
“You might want to send her a birthday card”.
You can use ‘may’ instead of ‘might’ here as there’s no real difference, but ‘might’ is more common.
To make a confession
We’ve all done it: eaten that packet of biscuits ourselves while a family member is out. English speakers use ‘may’ and ‘might’ to confess to doing something we shouldn’t have done:
“I may have eaten the whole packet of biscuits, sorry!”
“I might have eaten the whole packet of biscuits..”
Again, there’s no real difference between these two, because the word ‘may’ or ‘might’ is used to soften the bad news.