An Anti-Ballistic Missile is a missile used to intercept and destroy a ballistic missile. Perhaps in full it should be described as an "Anti-Ballistic-Missile Missile", but such an unwieldy name never really caught on.
Anti-Ballistic Missile systems come in two main varieties: tactical and strategic. This usually refers to the type of missile they are meant to intercept, with "tactical" and "strategic" being code words for "conventional" and "nuclear". ABM systems can also have warheads that are either conventional or nuclear, and that would usually coincide with whether they are used in a tactical or strategic role.
An Anti-Ballistic Missile system is really not different in concept from any other Surface to Air Missile system, since it is meant to fly from the ground and hit a moving target. However, the execution is much different, because ballistic missiles are usually flying much faster than even a fighter using afterburners. For this reason, the task of making an ABM system that can successfully hit a missile in flight has been likened to "hitting a bullet with a bullet". Despite this, there has been some success in overcoming the technical challenges, and some feasible ABM systems have been developed.
While ballistic missiles were in use since World War II, with the German V-2 rockets, the development of ABM technology was only even considered seriously in the 1960s, because radar, computer and aerospace components needed to be developed greatly before a usable ABM could be built. Even then, the system would be very expensive and might only defend against a small fraction of the incoming missiles.
The development and deployment of Anti-Ballistic Missiles is as much, if not more, about politics than about technology. During the cold war, the United States and the Soviet Union were in an unparalleled arms race, and the addition of ABM systems to the mix threatened to send the arms race into an even more vigorous leg. The reason for this is simple: The best thing to do if you suspect your opponent has something to shoot down your nuclear missiles is to build more nuclear missiles. And its easier to build a nuclear missile than to build an anti-ballistic missile. A nuclear missile is traveling very, very fast and is a comparatively small target. Building a missile to hit one is hard. On the other hand, a nuclear missile just has to hit the ground, which is very large, and also isn't moving. And, with a big enough payload, it doesn't even have to be too accurate. So if a nation wants to overcome an enemy's expensive ABM system, they can just churn out more relatively cheap nuclear weapons. They can also play some other tricks, such as adding multiple warheads to each weapon, or deploying electronic countermeasures and decoys. For this reason, both of the superpowers tried to limit the deployment of these systems, in a series of treaties, SALT I and SALT II, that spanned ten years and which involved so many twists and turns of domestic and foreign politics that The Byzantine Empire at its height would have looked like an election for girl scout treasurer. The politics of missile systems is still an issue, with America's (now suspended) plans of building an ABM system in Eastern Europe being a point of contention between the US and Russia.
To return to a technical issue, big missiles are only one way to deliver nuclear weapons. Along with the ICBM, nuclear weapons can also be fitted on cruise missiles, free fall bombs, artillery shells, depth charges, and even be transported by teams of a few people. (Although many of those methods only work with smaller, tactical nuclear weapons. Although "smaller" is a relative term with nuclear weapons.) These also are assuming a somewhat conventional form of nuclear warfare: a nation could also, for example, put a nuclear weapon on a freighter and set it off in an enemy port. In many cases, ABM installations would be one of the first targets to be attacked in such a war, after which larger weapons could be used. In other words, nuclear weapons are pretty hard to defend against.
So why have ABM systems been such a politically hot topic for the superpowers, despite the fact that they are expensive, not proven to work, and easy to destroy by a truly determined opponent? In what I am going to offer as a serious answer, although some people might not believe it, I think that the threat of really big nuclear missiles, screaming across the continent and invading our space, disturbs people's masculinity. I truly believe that missiles are feared by many because of their phallic overtones, and the deployment of an ABM system is motivated by something much more important than the desire to protect against the hellish death and misery that a nuclear weapon strike would lead to: it is designed to keep a man's penis from touching us. I know that I am not the first person to wonder about the sexual overtones of nuclear war, but it honestly seems to me to be the best explanation for the faith put in strategic Anti-Ballistic Missile systems.