MX was the provisional name given to the project to develop a new inter-continental ballistic missile when the start signal was given in 1972: "Missile-X" or "Missile, eXperimental". In 1982 with development nearing completion, the new missile was given the designation LGM-118A Peacekeeper, but the old name remains better known thanks to the controversy and debate that surrounded the project.
The origin of the MX project was a new generation of Soviet ICBMs that was expected to be introduced during the first half of the 80s. These would be accurate enough to effectively destroy American ICBMs in their silos, leading to a fear of a so-called disarming first strike from the USSR. It was decided to develop a new missile with increased accuracy, range, and mobility to meet the threat.
However, if you believe that the enemy is planning a counterforce strike against your missiles on the ground, and you require that they survives that attack, then deploying a heavy, MIRVed ballistic missile raises a problem. Since the missile would carry ten independent reentry vehicles (RVs), the USSR could theoretically justify directing ten RVs at each MX missile. So some kind of missile basing was needed that could not be destroyed by less than ten nuclear bombs.
Various exotic proposals were made, for example to base the MX missiles on airplanes (an airborne launch was demonstrated), or on small submarines off the continental socket, or in silos defended by anti-missile missiles. One of the more ingenious ideas was the "Multiple Protective Shelter" (MPS) system, where ten identical silos would be built for each missile. The missile would then be continually swapped around with nine decoys, so an attacker would have to destroy all ten silos to be sure to hit it. The idea failed due to the huge areas of land that would have to be dedicated to ICBM fields.
The basing problem was becoming all the more desperate as the missile itself was getting closer to completion. Some weaker (but cheaper) proposals were starting to look good - in particular the "dense pack" scheme. It called for grouping several nuclear-hardened silos only a few miles from each other in the hope that when one of them was destroyed the explosion would disturb RVs that were aimed at the others (the so-called fratricide effect).
In 1983, when everyone was thoroughly fed up and no satisfactory solution was in sight, the Reagan administration decided to simply place the first batch of the new missiles in the old Minuteman silos. There they were exactly as vulnerable as the Minuteman missiles they were replacing, but at least they were being deployed. In 1986 the first four missiles were delivered, and in 1988 a full squadron of 50 missiles was in operation.
For the second batch, a railway based system was being developed. (This, incidentally, was the solution the USSR choose for their comparable SS-24 Scalpel). The missile trains would be kept on alert in a large depot. At the warning of an attack, they would be rolled out and spread over a large area which the attacker would have to saturate with bombs to be sure to destroy all of them.
At the same time, a new missile program was started which solved the fundamental problem: the new missile was small, relatively inexpensive and re-used the Mk-21 RV developed for MX, and was to be road-mobile using a special nuclear-hardened tractor and launcher. Most importantly it only carried a single RV, so there was no need for it to survive an absurd number of direct nuclear hits. It was offically called the MGM-134A Small Intercontinental Ballistic Missile, but was nicknamed "Midgetman" in a joking reference to the Minuteman missile.
Despite the inherent survivability in their design, the railway garrison and the Midgetman both failed to overcome the final threat to their existence: the end of the cold war. They were cancelled in 1992. The 50 Peacekeepers that were in active service eventually suffered the same fate; the last one was taken off alert status in 2005, and they are currently used to launch satellites.