First, a disclaimer: a full general war with North Korea is incredibly unlikely at this point. North Korea would have to be suicidal to intentionally enter into such a war. Still, a sneakier, smaller attack may be likely as a face-saving measure by young Kim Jong-Un.
And what if that attack were to cause some sort of mis-communication, or the responding South Korean commander was a bit overzealous? Or if it inadvertently did more damage than anticipated and South Korea was forced by domestic politics to respond with war? How about if some idiots online decide to hack the most unstable dictator on the planet, and he is sufficiently shamed to consider an attack in retaliation to retain his domestic credibility?
So, just to play around with a hypothetical, what would happen if North and South Korea were to go to full war in the next year?
North Korea probably lacks the capability to attack Guam or Hawaii, and our missile defense systems would be prepared if they did. That restricts the theater of war to the Korean Peninsula and Japan.
Given the restraint the South has shown in the past, the North would probably be the first to strike. They have 1.1 million soldiers at their disposal, and an impressive array of medium-range missiles pointed south. That is almost double the South Korean forces, and an overwhelming advantage on the mere 28,000 U.S. troops on the ground.
Seoul would be an obvious first target of shelling. The North would use sleeper agents to mark targets for the artillery.
That shelling would then be stopped as the substantial U.S. Air Force presence in East Asia (easily outpacing North Korea's air force by a wide margin) would immediately target those missile arrays for destruction.
The next target would be North Korea's air defense systems and communications infrastructure, freeing our bombers and fighters to operate over North Korea with impunity.
The North's next step would be to initiate a tank blitzkrieg over the DMZ, while they insert special forces along the southern coast. These surprisingly capable soldiers would be deployed using submarines, to avoid the superior South Korean and American navies. The tanks, however, are mostly Cold War-era soviet ones, and would be easily destroyed in precision airstrikes.
Meanwhile, the air assault from America would continue, targeting bridges and key roads to shut down Northern freedom of movement.
North Korea would lose its momentum in the attack in as little as three days, as the U.S. coalition forces paralyze the entire Northern military, possibly without ever needing to step foot on Northern soil.
This would hopefully cause the North to back down and return to the armistice. Due to domestic audience costs (a nice way of saying Kim Jong-Un would face the serious risk of a coup) resulting from losing a war, though, they may not. If that is the case and the North continues to fight on, it would turn very bloody.
We would certainly win eventually, but only after "World War I levels of casualties."