Can and may belong to a category of verbs variously referred to as auxiliary, helping, modal, and defective. They are linguistic fossils, deriving from Old English conjugations that have dwindled through time to only one or two forms.
May and its past form might come from OE magan, “may, to be able.” In modern English, may sometimes carries the sense of expressing permission. Some parents still teach their children to make requests with the word may rather than can. The routine goes like this:
Child: Mother, can I play outside?
Mother: I’m sure you can play outside. The question is, “May you?”
Child: May I play outside?
Mother: Yes, you may.
In present tense, may and might are almost interchangeable. A subtle difference is that may can indicate a more likely possibility than might. For example, consider the following sentences:
“I may go to Billy’s game.”
“I might go to Billy’s game.”