Origins
People often say that Korea is America's forgotten war and there are many reasons to want to forget it. Over 42,000 Americans were killed or MIA and over twice that number were mutilated, all so that at the end of the war the border between the two Koreas could shift slightly. Korea was the first indication that during the Cold War, armed conflict would be more about superpower credibility and political solutions than military victory; that the United States was limited in dealing with even a small peninsular like Korea; and that its military might was not infallible.
The Korean War was, first and foremost, a Korean civil war. Korea was occupied by the Japanese Empire in 1905 and brutally exploited to aid the Japanese war effort, causing deep divisions between the collaborators and the resisters. Then, at the end of World War II, the country was split into two between the Soviet Union and the United States for the ostensible purpose of administering the surrender of Japanese forces. The country was supposed to hold elections on reunification under a United Nations mandate, but the Soviets refused to hold polls in their sector.
This was an early indication in the post-war world that the operation of the UN depended ultimately on the co-operation of powerful member-states. U.S. intervention would eventually take place under a UN mandate and with contributions from many other countries, a situation that was allowed to come about because the Soviets were boycotting the UN and so could not exercise their veto. They would not make the same mistake in the future.
Neither the North or the South's leaders wanted the division to stay permanent, and both plotted in the early years to see the peninsular reunified by force. In Seoul and Pyongyang, the Soviets and the Americans had to restrain their allies. Eventually, after repeated provocations by the South - which hoped to spark an invasion by the North and then use it as an excuse to conquer the whole country - the North rolled over the border in 1950. The invasion had finally received Soviet approval after U.S. Secretary of State Dean Acheson had said in a speech that Korea was not part of the U.S. "defensive perimeter" in Asia, implying that the U.S. would not intervene to protect South Korea.
This was later seen as a disastrous move as it invited the invasion and did not even turn out to be true. Meetings of the National Security Council in Washington did not even take time to cover whether or not an intervention was necessary, but simply took it as a given. The invasion was seen as part of a global Communist plot, driven by the Soviet Union, to test the United States. It was commonly seen as the breaking down of the post-war settlement and could easily have been interpreted as the first shot in World War III. And, if things had turned out differently, it might well have been. However, the decision to escalate always lay with the United States or China, as the Soviets actually provided very little direct support to the North; Soviet forces, of whom there were few, only rarely and accidently clashed with Americans during the war, and Soviet arms shipments to the North did not escalate markedly when the war began.
Course
North Korea is a country that is often - and rightly - seen as a basket case, but its military has been first-class from the start. Tens of thousands of North Koreans had fought in the Chinese Civil War and become adept at guerilla warfare, whereas its armoured units were modelled on the Soviet example and carried out an effective blitzkrieg in the South. American forces rushed to the defence of their ally but suffered a number of embarrassing defeats at the hands of the North, until eventually all the Americans and South Koreans had left was a small sliver of land around the city of Pusan in the southeast of the peninsular. The U.S. then carried out a brilliant amphibious landing behind enemy lines at Inchon, and the course of the conflict was reversed.
While the defence of South Korea had been the original goal, the U.S. now switched to a strategy of trying to unify the whole of Korea under the Southern government. The North's forces collapsed under the renewed American onslaught, but very few of them were actually being defeated in battle; most were carrying out what the North called the "great strategic retreat", a process that continued right up until the mountainous redoubt on the Chinese border. They knew something that the Americans, who followed them gleefully to the top of the peninsular, didn't. As soon as U.S. forces crossed the border between the two Koreas, the Chinese had taken the decision to intervene in force.
The Chinese decision to intervene had the potential to escalate the war into the ever-elusive World War III, but it was carried out primarily for Chinese reasons rather than because of a global Communist plot. The Chinese Communist regime had not yet solidified its control in China itself, and worried that a heavy American presence on its border could be used for counter-revolutionary purposes. The Chinese also felt a great deal of fraternal sympathy for the North Koreans, many of whom had fought in the Chinese Civil War and helped the Communists achieve victory; Chinese intervention in Korea hence made sense for ideological and material reasons.
The American and South Korean forces again collapsed under the Chinese offensive, until the fighting stabilized around the border between the two Koreas; the South Korean capital Seoul changed hands twice during this period. After the front had stabilized, several years of positional warfare ensued which was reminiscent of the trench warfare of World War I. After a while it appeared that neither side could achieve final victory and a ceasefire was signed at Panmunjom, solidifying the division of the country for decades amid an armed truce on the border which broke down - with heavy casualties on both sides - on a yearly basis.
Controversies
A number of episodes during the war were highly controversial. Firstly, both sides carried out massacres of civilians who they suspected of being their ideological opponents, and there have been unsubstantiated claims that American forces participated in these massacres. What is certainly clear is that the Northern strategy of total war, in which every man, woman and child was expected to take up arms against their enemies, resulted in arms being taken up against these civilians in retaliation. This was not only true on the ground but also during the air war, in which the United States Air Force demolished every modern building in the North during the period of the war, including strikes on the country's dams that flooded whole areas. Two million Koreans died on both sides in the conflict.
Then there was the role of the American general, Douglas MacArthur. MacArthur was a hugely popular figure in the United States and the closest the American Republic has ever had to a Caesar. Massively insubordinate and not unwilling to appeal to the American people and their elected representatives in the legislature over the head of the civilian leadership of the military, MacArthur was also a proponent of what was known as "rollback". He wanted to expand the war into China, drop nuclear bombs on Chinese cities, and try to finally "settle" the question of Communism in East Asia; such extreme action seemed at least plausible so soon after World War II, and the bombing of Hiroshima and Nagasaki. At one point he recommended creating a belt of radiation through China using Cobalt-60 to permanently separate Korea from China.
President Truman eventually fired MacArthur for insubordination, but he returned to America to be greeted by jubilant crowds and a pliant Congress. Thankfully, the danger to the Republic was headed off as the rest of the military establishment rallied around the President and declared MacArthur's plans to be unworkable. The most convincing argument against him was that opening up such a massive offensive operation in Asia would not only decisively lose any moral advantage the United States might possess in the battle against Communism, but also invite a Soviet invasion of Europe. Hence, the superpower conflict that had driven U.S. involvement in this far away land and imbued it with such importance contributed to restraining its escalation, a pattern that would be repeated in the decades to come.
Further reading
Bruce Cumings, Korea's Place in the Sun: A Modern History and Don Oberdorfer, The Two Koreas: A Contemporary History provide the best overviews of modern Korea, including the war.