first, where is Korea?
Korea is sharing a border with China. It is close to Russia, China, and Japan.
when, why, and how did the war break out?
The reasons why wars in the world break out are really intertwined and complicated. It is related to the economy, ideology, religion, and so many other things between at least 2 countries. My perspective will be one sided. So if you are interested in the topic, I recommend you to read more about it.
The Korean War was held in 1950-1953. Before we talk about the Korean war, it is important to understand that Korea was colonized by Japan from 1910 to 1945. When Japan was defeated in World War II, the United States and the Soviet Union divided Korea, which was under Japanese colonial rule. In 1945, the Soviet Union administered the north area of the 38th parallel and the United States administered south of the 38th parallel under “trusteeship”.
The trusteeship was determined in the Moscow Conference(1945) of Foreign Ministers, which was held in 1945 december. However, the countries who joined the conference made a decision without a single member from Korea attending this meeting1.
In the paper record of Moscow Conference 1945, the section on Korea consisted of four paragraphs. The third paragraph called for the establishment of a Joint Commission, under the control of a consortium of the United States, Soviet Union, United Kingdom, and China, including the decision that a four-power trusteeship of up to five years would be needed before Korea attained independence. Both the political left and political right of Korea opposed this trusteeship plan, with it being suggested that such a plan made Korea a vassal of the four powers.2
Koreans were traumatized by colonialism because of Japan. So many people were afraid that trusteeship would cause another period of colonialism. So there were many protests against trusteeship. The U.S. and Soviet Union tried to make a US-Soviet Joint Commission. But the problem was that the U.S. wanted to approve only the political party that is pro-america. And the Soviet Union wanted to approve the party that agreed on trusteeship. Additionally, within the Korean peninsula, there was a divide between those who opposed and those who favored trusteeship rather than the unification of North and South Korea, and this rapidly developed into a conflict between left and right.
As the trusteeship period came to an end, the United States transferred the Korean Peninsula issue to the United Nations. The United Nations initially decided to hold elections in North and South Korea to establish one united government. However, the Soviet Union refused to allow the UN delegation to come north of the 38th parallel(the UN was composed of pro-US countries). Because of this, it became impossible to hold general elections both in North and South Korea under UN supervision. So the UN General Assembly eventually decided to hold elections only in South Korea that can be monitored. However, this later became a decisive cause of the division between South and North. Because “officially” the South held elections by itself and the North held elections within itself.
Here we need to look a little more into the creation and execution of the United Nations. This is because most people think of the UN as “neutral” but it was(and is) not.
1. Kang Man-gil , 『 Our history in the 20th century』 ,2009, <How did the Korean War occur?>
2. Moscow Conference (1945) - Wikipedia