Use can to express ability, as in I can carry those books. Both can and may can ask for or give permission. Can I borrow this book? is ordinary everyday English. May I borrow this book? sounds more formal.
Read the sentence in context
- Ability. Jules can read this in Spanish. The sentence describes what Jules is able to do.
- Permission. You may keep the book for two weeks. The speaker allows it. You can keep the book for two weeks can grant the same permission.
- Possibility. The library may close early today. The sentence refers to one possible event. The library can get crowded on Saturdays describes a general possibility.
Match the word to your purpose
If you’re asking permission, choose the register that fits the situation. If you mean that a specific event might happen, see may vs. might for more examples.
