The Catholic Church, along with some other Christian churches, recognizes that certain people are paragons of holiness and worthy of emulation. Through a complex process of investigation and testing, the Church may determine that a particular individual is holy enough to be called a "saint".
A saint is not free of sin, nor above the need to confess, receive Holy Communion and practice their faith both publicly and privately. Saints have their faults and their trials. The Church recognizes them for having used the gifts that God has given them to overcome their humanity and be closer to the example for us all that is Jesus Christ.
One of the oddities around the lives and deaths of certain saints is the appearance that their lifeless body does not decay as would a normal person's. This is not embalming, but an apparent halting of the natural process that returns man to dust. While some of these saints remain non-decayed to this day, others decayed after several generations of incorruptibility.
Archaeologists have found human bodies in remarkable states of preservation in varied places throughout the world. Nearly all were the result of embalming or the extreme dryness of the local climate.
While most of the reports of incorruptibility date from before the Reformation, the exhumation of the body of Pope John XXIII, of Vatican Two fame, revealed a body remarkably free of decay.
The Church takes no official position on these mysteries. There are bodies of non-Christians, and of those whose lives were less than holy, that have been found to display incorruptibility. Still, believers in the Church have long equated incorruptibility with holiness, and some of those saints who have displayed this trait have done so in spite of attempts to affect the process. More than one saint, John of the Cross is a good example, has been covered in lime to encourage decay and yet their bodies have not been dissolved as one would expect from the chemical action of the lime.
Incorruptibility is as much a spiritual issue as a physical one. If the incorruptibility of a saint's body allows a believer to see his holiness and example and to practice the qualities that made the saint holy, then the incorruptibility is an asset to the growth of the faith of that believer. None of the teaching of Christ are denied by apparent incorruptibility nor is belief in incorruptibility required for salvation or lawful participation in the sacraments of the Church.
Incorruptibility remains a mystery, a mystery that may display the holiness of a saint and may bring a believer closer to Christ.