There probably are several ways.
If you are saying "I have the ability," then it is ~ㄹ 수 있다.
Ex: I can speak Korean: 한국어 할 수 있어요. (or "한국어 할 수 있습니다" or in 반말: 한국어 할 수 있어.)
For the negative, you can say: "할 수 없어요."
if you are implying that you want to be able to, but are unable, you could say, "못 해요."
If you are asking "Is it allowed to do that?" or saying "It is allowed to do that," then 어도 되다. However; the negative (saying it is NOT allowed) is 면 안 되다. This is more like "May" than "can," but native English speakers (at least the ones from the US) interchange "May" and "can" so often that I thought it good to include this.
Example, when someone asks if they can pet our dog, they ask, "강아지를 만져도 돼요?"
To answer telling them they can pet him, we say "만져도 돼요."
If we were to tell them they cannot pet the dog, the answer would be, "만져면 안 돼요."
Others can give other answers, too, I'm sure. :)