What is the local name for Korea? South Koreans will answer: Hanguk. It is partially true. The official local name for South Korea is Daehan Minguk and it is shortly called Hanguk. But North Korea's name is "Chosun Minjujuui Inmin Konghwaguk" and North Koreans call Korea "Chosun (Joseon)" (or "Konghwaguk", the Republic). North Koreans never use "Han" and South Koreans do not use "Chosun" with few exceptions such as "Chosun Ilbo". The local name for the Korean Peninsula is "Han Bando" in the South and "Choson Bando" in the North. Hangugeo and Hangungmal refer to the Korean language in the South. Chosuno (Joseoneo) and Chosunmal (Jeosonmal) in the North. Hangul is called Chosongul in the North. Only "urimal" (our language) is used in both countries.
Historically speaking, Chosun was an old kingdom in the northwestern part of the peninsula, which was destroyed by the Chinese Han Empire in 108 BC. It was the name called by the Chinese and they probably called themselves differently. In contrast, Han was a region in the southern part of the peninsula, which was also called by the Chinese. The Chosun Dynasty, also called the Yi Dynasty, was the kingdom which ruled the whole peninsula (1392-1894). The founder Yi Seonggye adopt the name Chosun after the old kingdom. After the independence from the Manchu Empire, the kingdom was renamed to the Han Empire (Daehan Jeguk) in 1894. When Korea was annexed to Japan in 1910, Han was renamed to Chosen (Japanese pronunciation of Chosun) again. In 1948 when the Korean Peninsula was divided and South Korea and North Korea were established, the South adopted Han and the North adopted Chosun.
Thus the both county consistently use the different names, but a problem was occurred in Japan. In Japan, South Korea is called Kankoku (Hanguk) and North Korea Kita-Chosen (North Chosun). Then North Koreans in Japan said in remonstrance that it was the discrimination against North Korea not to use what North Koreans call themselves, and they tried to force Japanese media to call it "Kyowakoku" (the Republic). They made a compromise and Japanese media call it "Kita-Chosen, Chosen Minshushugi Jinmin Kyowakoku" (North Chosun, Democratic People's Republic of Korea) when it is appeared at the first time. Of course, most Japanese are dissatisfied with the long-winded name.
If we consider Korean Russians, things are more complicated. Korean Russians call themselves "Koryo people", not Chosun or Han. Koryo was a kingdom followed by Chosun Dynasty and became the etymology of the English name "Korea". They adopted the name to avoid the North-South conflict. Koreans cannot unify even their nation's name.