The Vietnam War was the second of the two major proxy wars fought during the Cold War, and is the one most remembered in the United States due to the widespread protests that took place during it, along with it marking the first time the US had unequivocally lost a war. It was fought between the Soviet-backed North Vietnamese and the US-backed South Vietnamese. The war ended with the single largest helicopter evacuation in history as the US forces pulled out of Saigon, leaving behind their erstwhile allies.
Vietnam was originally a French colony, conquered during the 19th century along with the rest of French Indochina. During World War Two it was occupied by the Japanese as part of their "Greater East Asia Co-Prosperity Sphere." After the Japanese were defeated, the Vietnamese, mainly the Communist revolutionary group known as the Viet Minh, were unwilling to go from one foreign ruler to another and chose to continue their fight for independence, this time against the French.
Though during the war the Viet Minh had enjoyed the support of China, the USSR, and the United States, once the war was over the USA ceased its support and returned to its old alliances. Despite this setback, the Viet Minh were able to eventually force France to withdraw, but this led to worse problems on the horizon. Like Korea before it, Vietnam was separated into two parts, one communist, the other capitalist. And like Korea before it, this resulted in war.
Vietnam was in many was the second Korea, but there were two key factors that made it different. First was that Vietnam had few remaining native Type-II individuals, as France had impressed all that it could find during World War I, and then sent them to their deaths at the battle of Verdun. This proved to not be insurmountable for the Soviet-backed North Vietnamese who were being supplied mainly with conventional technology, including the soon to be infamous AK-47, and thus did not require wizards in their ranks for maximum combat effectiveness. The US, however, found their expertise with and reliance on magi-tec to be a crippling weakness when combined with the near total lack of native Areum wielders, as it meant that the US had to send their own mages into every battle that they wished to bring their advantage to bear in. Secondly, perhaps due to the resentment this caused, or perhaps due to the unfortunate fact that the leader they were supporting was a despot at best, the US troops failed to "win the hearts and minds" of the native Vietnamese, and thus never enjoyed the kind of popular support they had possessed in Korea. This resulted in the impressive effectiveness of the Viet Cong insurgency, which fostered even greater resentment for the Vietnamese both in the US troops and among the US populace.
Eventually, the US populace would decide that they had sacrificed enough in this incredibly unpopular war, and would force the US president to concede, and withdraw his support from the South Vietnamese, attempting to "Vietnamize" the war by training native troops to take the place of the departing US ones. This failed, and the American departure turned into a last minute evacuation as northern forces advanced into Saigon, thus bringing the war to a close.
Vietnam was the first war in US history that it had unequivocally lost. The war of 1812 had been a stalemate, roughly, while every other war since then had resulted in the US being on the side of the victors. The war thus had a massive impact on the American psyche, and according to some contributed to the cooling of the Cold War afterward due to the increasingly powerful peace movements. Vietnam itself became a full-fledged Communist country and remains one to this day, acting as a living rebuke and embarrassment to the United States, more than Korea ever was.
The Vietnam war was in many was a testing ground for the two powers, a place where they could field test their new weapons and technology against the other without actually coming into direct conflict. The Soviet victory despite US magical superiority served as a strong reinforcement to the USSR's military doctrine and called the US's own doctrine into question. US military analysts correctly pointed out that the issues were unique to the area and situation, and were unlikely to be repeated in a direct conflict between the two powers. Thus, both sides wound up adhering even closer to their chosen field of expertise, a trend that continues to this day.