to express ability (in the sense of being able to do something or knowing how to do something):
He can speak Spanish but he can't write it very well.
to expression permission (in the sense of being allowed or permitted to do something):
Can I talk to my friends in the library waiting room? (Note that can is less formal than may. Also, some writers will object to the use of can in this context.)
to express theoretical possibility:
American automobile makers can make better cars if they think there's a profit in it.