The Korean Conflict/Korean War (KC or KW) was the first major-scale battle since World War II, some 10 years prior. It was also the first war in which the United Nations, as an organization, sponsored war. By this this time, the use and principles of nuclear weapons has been well established primarily due to the use of two nuclear devices by the United States on Japan in the early 1940's. However, even though the capabilities existed, no one used them during this conflict. I presume here to try to examine the reasons why.
The KW was anachronistic in many ways. Although there had been many (however minor) advancements in military technology, it was fought primarily with older WWII era weaponry. Researchers have many viewpoints on this, most of which fall into three camps: 1) The UN forces did not want to invest capital in what they felt was going to be a minor and short-lived conflict, 2) with the advent of the atom bomb, ground warfare was obsolete, or 3) the more debatable: there was a purposeful effort to keep the conflict from escalating into World War III.
From the USA's perspective, the second seems the most likely. The Secretaries of Defense Harry Truman (the US president at the begining) had appointed, James Forrestal and Louis Johnson, not only stopped development of new ground infantry arms and communications, but forced a complete change in Army training methods, shaping it to produce combat-trained civilians, rather than professional soldiers prepared to face combat and death. A lower premium was placed on the 'outdated' ground troops as more advanced techniques such as air superiority and nuclear warfare began to take hold.
But this still doesn't answer the question of why nuclear weapons were not employed to end the conflict.
The majority of KW analysts agree that, had the US employed a small number of strategic nuclear attacks and had North Korea been a solitary force, the aggressor forces would surrender, not due to the attacks themselves, but due to the mindset that once the US uses 'nukes' once, they won't hesitate to use them again. The US had a huge stockpile of low- and high-yield nuclear weapons in its arsenal that is had the capability to exterminate North-Korea's offensive capabilities.
As most people know, nuclear weapons are devastating long after the initial blast due to nuclear fallout, the radiation left after a nuclear blast. Although 'environmentalism' as we know it today really didn't exists in the 1950's, there are still worry that the fallout could reach the high atmosphere and cover the entire globe, harming the very people nuclear weapons were mean to be protecting. The sheer number of devices planned to be used (not only on the Korean peninsula, but other communist powers) was impressive: 20 in Moscow-Gorki, 12 in Leningrad, 52 in Volga and Donets Basin, 15 in Caucasus, 15 in Vladivostok and Irkutsk areas, for a total of over 110 devices.
Another, and more likely reason is that North Korea, China and the USSR were in a strategic alliance. Although China and North Korea themselves had no appreciable nuclear capability (that we know of), the USSR had an unknown (but presumably sizeable) nuclear capability. Here is where the concept of 'mutually assured destruction' comes in to play. If the US nuked North Korea, the Soviets would, almost certainly, retaliate and so the US would have to simultaneously attack the USSR. Of course, China would then join the conflict and there would be a 3 nations (USSR, China and N. Korea), 2 of which are among the largest in the world in size and population, in a full-scale nuclear battle with the UN and, more specifically, The United States. However, many of the other key UN nations (Britain and France specifically) were strongly opposed to the use of nuclear weapons. Going on this, there was a real chance that they would convince the other UN nations to leave the US to it's fate, leaving the US to fight the Asian aggressors themselves, a battle that would be difficult (if even possible) to win.
As it was however, there seems to be a strong belief among war historians that then President Eisenhower had been convinced by his generals to employ nuclear weapons had the talks at Panmunjon failed. However, as you may well know, the talks succeeded and led to the Armistice at Panmunjon, and there was no nuclear exchange in 1953, and there has not been one since the initial use some 57 years ago (and counting).
My Opinion: If you can't tell, I'm bias toward the avoidance Mutual Assured Destruction reasoning. I feel that Eisenhower was smart enough to realize that the US couldn't take on the USSR-China-Korea forces on their own, especially not without sanctions from any other nation. I personally don't believe that Eisenhower would have ever used the US's nuclear capabilities, though I also feel he was not above throwing conventional weaponry (i.e. people) at conflict for years to come.
A challenge by Infinite Burn and KidTamago