Originally I think 한국 is meant to mean the whole of Korea, to South Koreans, but naturally they use it to mean South Korea. And when talking specifically of North Korea, they use 북한.
As for the official name of South Korea (The Republic of Korea) it is 대한민국. North Korea (The People's Democratic Repiblic of Korea), I don't remember, but it's very long. ㅋㅋㅋ
North Koreans on the other hand, call Korea 조선 and South Korea 남조선.
To any native Koreans, feel free to correct me, if there's any mistakes or I've misunderstood something~