Regarding ICBMs and interception:
There are basically three phases of flight: boost, midflight, terminal.
In the boost phase, the missile is a large, slow target with a hot rocket plume behind it. Furthermore, it's over enemy territory. It is easy to shoot down at this point, although the window of vulnerability is only about 15 seconds and it's typically deep inside enemy territory. Nevertheless, if a ballistic missile attack was anticipated, a penetrating stealth aircraft could potentially engage and destroy the missile in this phase.
In the midflight phase, it's moving fast and it's very high up. Ground-based defences probably could not hit it easily. This would require bullet-on-bullet precision from distances of 1200km for ICBMs (in suborbital spaceflight). Doing that with a kinetic missile would be near impossible, technology regardless. Getting the interceptor up there is difficult enough.
A space-based defence system has half a chance of stopping missiles in midflight. Its countermeasures do not have to fight gravity to reach their target, letting them expend more fuel on manoeuvering to hit it.
Ground-based lasers would be attenuated greatly by the atmosphere. They would have to be extremely powerful to shoot upward through miles of atmosphere and still bring down an ICBM, especially if the enemy decides to give them a reflective, antilaser coating. Space-based lasers would fare much better and would require less energy input to impart the same level of energy to the target, possibly enough to overcome antilaser measures.
Probably the best solution would be to use some sort of EMP or nuclear device in the high atmosphere to destroy multiple ICBMs simultaneously.
In the terminal phase, the warhead (it may no longer be a whole missile if the launch vehicle has MIRVed) is moving very fast, it's making final adjustments to its course, and it's over friendly territory. The goal, should it get this far, is primarily to stop it hitting its intended target. Diverting it by a few miles will still likely result in many deaths, but fewer than if it had reached the planned impact point.
Remember that modern nukes are fused for airburst, sometimes at thousands of metres altitude. Once it enters terminal flight, there is an uncertain amount of time available to the defender to destroy or divert the warhead. Diverting it will result result in a nuclear yield. Destroying it will not, but still scatter radioactive debris.
At this point kinetic missiles are almost certainly out of the picture unless the launcher is within the expected target zone. The warhead is supersonic. An airborne laser may do the job, but can only protect the airspace in front of it out to a limited range. Also, the fire rate of such a defence is limited; not only does it have to cool and recharge the laser after each shot, the shots themselves may require a substantial period of time to pierce the shell of the warhead's fuel tank. Keeping a laser focused on a supersonic target is not an easy task.
Ground-based emplacements are not limited by the mass, bulk and power requirements of their laser. These sites could be substantially more powerful and fire pulses capable of destroying warheads almost instantly. The tracking problem would be no more difficult that it is for an airborne platform.
Finally, the easiest and lowest-tech way to bring down nukes in the terminal phase is a high-explosive shrapnel missile of some sort. The tracking system need only be good enough to get close to the target before detonating. A larger explosion means that less accuracy is required, plus it may hit multiple warheads.
A laser would likely destroy an inbound warhead. An interceptor missile would destroy it on impact (very unlikely) or divert it with a nearby explosion.
The USA's missile defences currently consist of THAAD, PAC-3 and the experimental ABL. THAAD stands for Theatre High Altitude Air Defence, the Army's part of the missile defence programme. It is designed to intercept warheads at altitudes of 120 to 150km using kinetic hit-to-kill missiles. Despite the difficulty of hitting a ballistic missile in the midflight phase, THAAD has demonstrated this capability, although I'm not sure how reliable it would be against a real nuclear attack.
PAC-3 (Patriot Advanced Capability 3) missiles are capable of intercepting ballistic warheads at ranges of about 10km. These are absolute last-chance defences; they would probably have some degree of success mainly due to quantity (put enough explosive in the air and you're bound to hit something) but the PAC-1 and PAC-2 upgrades also provided facility for attacking specifically ballistic missiles.
That's the background and should answer any more questions people have about missile defence and the difficulties involved. If I've got anything wrong, please correct me. Some figures came from Wikipedia and may not be accurate.
