I covered some of this in the last thread on North Korea (when they made their last tests), but I'll talk about some of it here.
The Nuclear CapabilitySimply put: if you're not in Japan or South Korea, don't worry.
The North Koreans don't have a great launch capability. They managed a single success with a rocket carrying a "satellite" (NASA confirmed that
something is in orbit, they just don't know exactly what), after a long string of violent, fiery failures. Just because a single rocket succeeds doesn't mean that they've achieved 100% success rate. Indeed, it would be surprising for them to manage regular success with ICBM launches at this point in their development. Launching a satellite into orbit and letting the rocket burn up in the atmosphere isn't the same thing as directing said rocket to land on a specific target and have the nuclear device on the nose detonate. There's a good possibility that their missiles would miss entirely, or that the nuke would fail to go off (How many devices have they tested that failed to do anything as opposed to the three or four that did?). Assuming the rocket didn't just topple over during launch and explode against the pad.
Our Defense CapabilityThe reason I said only South Korea and Japan really need to worry is because they're the ones that they hate who are close enough that they may not be able to launch an adequate defense. The United States (their next target after South Korea) is not only much farther away, but is armed with various missile interceptors and other defenses. The only areas that they could competently hit right now (if the theories of them using a miniaturized Pakistani design is true) are Alaska and our Pacific islands like Hawaii and Guam. Someone in California won't have to worry much.
Even then, the American ABM system is virtually catered to the threat of a nuclear launch from the east. This is a quote from another site regarding such, which has been backed by a longtime military analyst:
We have 30 operational GMDs right now. Using a standard look-look, shoot-shoot, look-look, shoot-shoot, that's enough for shooting down 7 incoming ICBMs with absolute confidence, 2 GMDs being in reserve of that requirement. Firing 2 interceptors at each missile instead of 4 would increase the shootdown to around 15 ICBMs, though with our confidence interval of a successful intercept falling toward about 98%. That still gives us a relatively large likelihood of no missiles at all getting through, and of the one that does hit failing to detonate or missing so widely as to not cause casualties (since the malfunction rate might have been as high as 33% for early nukes). Collectively that means that the DPRK needs greater than 20 ICBMs to have any realistic chance of doing damage to the US mainland, and if the objective is to overwhelm the system and punch through to hit 15 targets, you'd need around 60 missiles (since in a saturation attack we'd probably fire a single interceptor at each incoming ICBM for 80 - 90% hit rate). That would be enough to hit every major urban area on the west coast at least once, but a lot of the west coast is going to be out of range of this thing. On the other hand the warheads are so weak and the target cities so large that something like LA can soak like 20 of these warheads and still be a partially functional city the next day -- they still built stuff in Hamburg after the firebombing level, at least, is what I mean to say.
tl;dr We can probably shoot them down. The North Koreans would need to simultaneously launch dozens of missiles to reliably get through our screen.
The Ground War Assuming we don't respond simply by nuking or even conventionally bombing the DPRK into a scorch mark, a ground war on a scale not seen in Western nations for decades will follow. There's a good reason for the effort to use a nuclear deterrent: everyone in the NK military should know that they'd be crushed by modern numbers and technology, especially without China on their side. Nuclear weaponry is more than just the first choice. It's probably the first 20 choices before they'd resort to ground war.
Unfortunately, they'll probably be forced into that. And it won't be pretty for them.
The military of North Korea is, simply put, outdated and broken.
This is from 2004, but it's likely still relevant. I'll mention the important parts. About half of their equipment is 1960s vintage, and most of the rest is even older (this includes biplanes as improvised STOL scouting aircraft). There's also the issue of logistics: they already don't have enough spare parts and fuel to maintain their vehicle pool, and a war will stretch their supply lines even thinner as they need to actually USE all of that stuff for something more stressful and wasteful than simple training exercises, and all of their enemies know that one of the first things you target is supply lines. We probably won't even need to fight the majority of their arsenal, since they simply won't be able to maintain it. And there's not even a guarantee that they have enough trained pilots and vehicle crew to staff the ones that are operating. Their soldiers may number in the millions, but it consists mainly of unwilling conscripts being forced forward more by the threat of execution than a true desire to fight a war. They'll be armed mostly with AK and SKS pattern rifles, and probably rely on human wave attacks akin to the Korean War (surprise surprise). DPRK military doctrine is firmly stuck in Soviet era inflexibility, with high leaders delivering orders that must be followed to the letter. Individual initiative is discouraged, and soldiers must simply follow orders or be shot in the back by their own officers.
Meanwhile, the US and South Korea maintain a firmly modern military. The South Koreans have already quite plainly told the DPRK that they can and will fire cruise missiles directly at their governmental headquarters (which is a publicly known target; even Google Earth lets you look down on them), so we can assume that the first step in case of a war will be to launch right there. We'll be able to attain air superiority extremely quickly, as well as send in the navies to blockade the shore and provide extra firepower for ground operations.
On the ground side, we're individually superior. Modern assault rifles fitted with grenade launchers and various optical sighs, including night vision devices, are the standard rather than the exception. There are large numbers of designated marksmen and automatic riflemen, which ensures that at least at the squad level, there will be at least one scoped marksman's rifle and one light machine gun to provide precision kills and large suppression/massive firepower, respectively. An M249 or Daewoo K3 will certainly work wonders against those aforementioned mass rushes. Anti-tank weapons like the AT4, M72 LAW, and SMAW are tough enough to easily punch through the old armor used by the DPRK. I would predict a single AT4 would be enough to knock out one of their old T-55s, which is the most numerous tank in the military; considering that one AT4 can be carried on the back of a soldier on top of his normal combat load (and quite commonly is), this gives massive anti-tank capability to the NATO forces. The soldiers also carry heavier equipment loads that include body armor, personal radios, large quantities of ammunition (current US combat load is 270 rounds, or 9 rifle magazines), demolition gear, and various other special use equipment.
Most importantly, we have logistics. The US is so successful because it can supply massive amounts of food, fuel, spare parts, and ammunition overseas with ease. It's very likely that none of the NATO parties or South Korea will starve in the field, or want for ammunition outside of individual cases where a small group faces overwhelming firepower. Meanwhile, standard tactics dictates constant bombing of North Korean supply lines to stretch out their capabilities as thin as they can go. It's estimated that they maintain 100 days of emergency ammo and fuel for a "full scale war" and 500 days of food, but this is assuming that none of it gets captured or simply blown up by the invaders. Moreover, they maintain such a reserve because they spend very little fuel and ammunition in training; most of their soldiers and vehicle crews and pilots haven't had more than token live fire exercises, and their pilots have extremely low flight time hours. They just don't have any experience.
If anyone would like, I can compare individual pieces (like rifle vs. rifle or tank vs. tank) in more depth on request. Just because I enjoy talking about this shit.