We use May I …? / Could I …? / Can I …? to ask for permission. Bear in mind that «may» is formal and we use it when we don’t know the person very well. «Can» is, on the other hand, informal and is used to address a person whom we know well. «Could» is more polite that «can».
May I use your computer for a moment?
Can I ask you a personal question?
Could I borrow your pen?
Some other common phrases are:
Do you mind if I smoke?
Would you mind if I asked you something?
Is it okay if I sit here?
Would it be all right if I borrowed your car?
When someone asks for permission, we can reply in a number of ways: Yes, of course. / Of course. / Certainly. When we know the person very well, we can reply with / Go ahead. / No problem.