“Korea” is the modern spelling of “Corea”, a name attested in English as early as 1614. It is an exonym derived from Cauli, Marco Polo’s transcription of the Chinese 高麗 (simp. 高丽, MC Kawlej, mod. Gāolì). This was the Hanja for the Korean kingdom of Goryeo or Koryŏ (고려; 918–1392), which ruled most of the peninsula during the time of his travels. (Scholars who discount the historicity of Polo’s account instead derive it via Persian variations of the same Chinese name.) Goryeo’s name was a continuation of the earlier Goguryeo or Koguryŏ (고구려; 37 bc – ad 668), the northernmost of the Samguk (the Three Kingdoms of Korea), which was officially known by the shortened form Goryeo after the 5th-century reign of King Jangsu. The original name was a combination of the adjective go (“high, lofty”) with the name of a local Yemaek tribe, whose original name is thought to have been either *Guru (溝樓, “walled city,” inferred from some toponyms in Chinese historical documents) or *Gauri (가우리, “center”). With expanding British and American trade following the opening of Korea in the late 19th century, the spelling “Korea” appeared and gradually grew in popularity; its use in transcribing East Asian languages avoids the issues caused by the separate hard and soft Cs existing in English vocabulary derived from the Romance languages. The name Korea is now commonly used in English contexts by both North and South Korea.
In South Korea, Korea as a whole is referred to as Hanguk (한국, [haːnɡuk], lit. ”country of the Han”). The name references the Samhan—Ma, Jin, and Byeon—who preceded the Three Kingdoms in the southern and central end of the peninsula during the 1st centuries bc and ad. Although written in Hanja as 韓, 幹, or 刊, this Han has no relation to the Chinese place names or peoples who used those characters but was a phonetic transcription (OC: *Gar, MC Han or Gan) of a native Korean word that seems to have had the meaning “big” or “great”, particularly in reference to leaders. It has been tentatively linked with the title khan used by the nomads of Manchuria and Central Asia.
In North Korea, China, Japan, Vietnam, and Chinese and Vietnamese-speaking areas, Korea as a whole is referred to as Chosŏn (조선, Joseon, [tɕosʰʌn],, (朝鲜), Cháoxiǎn, (朝鮮), Chōsen, Triều Tiên (朝鮮) lit. ”[land of the] Morning Calm”). “Great Joseon” was the name of the kingdom ruled by the Joseon dynasty from 1393 until their declaration of the short-lived Great Korean Empire in 1897. King Taejo had named them for the earlier Kojoseon (고조선), who ruled northern Korea from its legendary prehistory until their conquest in 108 bc by China’s Han Empire. This go is the Hanja 古 and simply means “ancient” or “old”; it is a modern usage to distinguish the ancient Joseon from the later dynasty. Joseon itself is the modern Korean pronunciation of the Hanja 朝鮮 but it is unclear whether this was a transcription of a native Korean name (OC *T[r]awser, MC Trjewsjen) or a partial translation into Chinese of the Korean capital Asadal (아사달),  whose meaning has been reconstructed as “Morning Land” or “Mountain”.