|South Korean actors playing North Korean soldiers in a movie.|
The nation that we call South Korea in English refers to itself as "Hanguk" (한국). "Han" is the name of the Korean ethnic group (not to be confused with the Han Chinese ethnic majority; in Chinese, the tones on the two words are different, as well as the hanzi for those words). "Guk" is the Korean word for nation, state, kingdom, or country (cognate with Mandarin guó 國). The formal name of the South Korean state, often translated as the "Republic of Korea," is the "Daehan Minguk." "Dae" is cognate with Mandarin dà 大, meaning "great." "Min" is cognate with Mandarin mín 民 and means "people"; hence, "minguk"—a people's country—means "Republic." (Sometimes you see the full phrase in hanja, 大韓民國, on soldier's uniforms and equipment.)
North Korea, in contrast, calls itself "Joseonguk" (조선국). "Joseon" was the name of Korea's longest-lasting dynasty, which lasted roughly half a millennium from just before the beginning of the 15th century to the brink of the 20th; the Joseon Dynasty is what most people still consider the apex of "traditional" (that is, pre-modern) Korean culture. It was also the name of the quasi-mythical, almost pre-historic Korean ancestral state (sometimes called "Go-Joseon" or "Great Joseon"). North Korea has no better claim than South Korea to be the legitimate heirs of the Joseon Dynasty, which was destroyed by the Japanese Empire; it's simply a propagandistic move.
North Korea's formal name, usually rendered in English as the "Democratic People's Republic of Korea" (or "DPRK" for short), is the "Joseon Minju-jueui Inmin Gonghwaguk" (조선민주주의인민공화국). "Minju" means "democracy" (from min for "people," as noted above, plus "ju," cognate with Mandarin zhǔ 主, meaning "owner/master": hence, a system where the common people are the masters. "Jueui" is a principle, belief, or doctrine (cognate with Mandarin zhǔyì 主義)—which people who know Korean will recognize from other words like chaeshikjueui (채식주의, "vegetarianism") and 사회주의 (sahoejueui, "socialism")." "Inmin" is an intensified way of saying "the people," combining two words from Chinese with a similar meaning into one (cognate with Mandarin renmin, like in the word renminbi), while "gonghwaguk" is different way of rendering the word "republic" in Korean—with a connotation more like "commonwealth"—cognate with Chinese gònghéguó 共和國, "shared-harmony country."
Here's where things get interesting. Each Korean state refuses to use the other's name. What we call "North Korea" refers to its southern neighbor as "Nam-Joseon" or "South Joseon"; what we call South Korea, in turn, calls the state to the north "Buk-Han" or "North Han." (Buk and nam are the Korean words for north and south, cognate with Mandarin bei 北 and nan 南.) (Sometimes, when South Korean newspapers refer to North Korea, they just use the hanja for "the north," 北.)
One last thing. Why, you may wonder, do we in English call Korea "Korea" at all, when that doesn't bear any resemblance to what Koreans call themselves? The dynasty that preceded the Joseon called itself Goryeo (고려), and though by the time Europeans first came into contact with Korea in the late 16th century, the Joseon Dynasty had long held power, it seems that they got the impression that the place was called "Goryeo." I'm not totally sure why this should be the case—did the people themselves still refer to their identity as "Goryeo"? Did the Chinese still call them that? In any case, the name stuck—and given that in Korean, the letter "G" at the beginning of words sounds closer to a "K" sound, and the "eo" vowel would sound rather like the letter "a" to a wayward 16th-century Portuguese guy, "Korea" is actually not so far off base.