Even native English speakers often misuse "may."
If you're asking to enter someone's house, the proper phrase would be "May I come in?" But many people say "Can I come in?" instead. The difference is "may" asks for the homeowner's permission. "Can I come in" is literally asking "do I have the physical ability to enter?"
Another meaning of "may" is to express an uncertainty, or a possibility: "He may be angry with me, I'm not sure."
Or the pilot of an airplane might say: "The weather looks cloudy over the Atlantic, we may run into some turbulance."