Korea was a peninsula of Asia on Earth. Located between the Yellow Sea on the west and the East Sea on the east, the peninsula was separated from Japan by a narrow strait. The peninsula was home to two nations, and the northern nation shared a border with China.
Korea can be seen (barely visible) for a split second on a Pacific map flashed on the screen during "The Cage". Graphical enhancement of the image can make out the name "Korea" as well as a region which is intended to be the city of Seoul.
The term "Korea" can be applied to both the geographical region and the actual countries of North and South Korea. While the Korean peninsula was visible on the downloaded world map from "The Cage", the nation of Korea has never been mentioned in Star Trek.
Korea was also the site of the Korean War which has never been mentioned in Star Trek, although a Korean War service ribbon can be seen worn by a United States Air Force officer in "Tomorrow is Yesterday".