There's some interesting history to why Korea is called by two different names depending on which Korea you're referring to.
As some other answers have mentioned already, each Korea is referred to by different names.
Apologies for the spacing, emphasis isn't playing nicely with me today.
North Korea:
北{きた}朝鮮{ちょうせん} (Short for: 朝鮮{ちょうせん} 民主主義人民共和国{みんしゅしゅぎじんみんきょうわこく})
North Korea (Short for: Democratic People's Republic of Korea)
This name is derived from the 조선 (朝鮮, Joseon) dynasty (the first one, often referred to as Gojoseon).
South Korea:
韓国{かんこく} (Short for: 大{だい} 韓{かん} 民{みん} 国{こく})
South Korea (Short for: Republic of Korea)
This name is derived from 한국 (韓國, Hanguk) name that South Korean people call themselves (people of "Han", or, the Han dynasty that defeated the Gojoseon dynasty that resulted in the 삼한 (三韓, Samhan), the three kingdoms of ancient Korea)
Unified Korea:
高麗{こうらい}
This name comes from the 고려 (高麗, Goryeo) name that Koreans gave their unified nation during the Common era of Korea.