Historically this refers to legalization/legitimization of Christianity by the Roman Emperor Constantine I in the early 4th century. The persecutions ceased, which of course was a good thing for the church, but the policy reversal, encouraging Christianity and beginning to fuse it with the state led to:
1. A watering down of Christianity. The church begins to lose its identity, its sense of salt and a spirit of nominalism began to creep in. Almost all histories agree on this fact, even someone as mainstream and ecumenical as Schaff.
2. A loss of identity, the sense of antithesis, in turn leading to new deviations of pietistic expression. Monasticism flourished from this time forward. Many innovations and Biblical deviations had already begun to make headway, but from this point they grew rapidly. The veneration of martyrs and their relics now expanded to a host of new corruptions.
Theologically I'm arguing for the sufficiency of Scripture as per 2 Timothy 3. Scripture is sufficient for the life and doctrine of the Church. Though the issue of cannon had not been quite settled, in practice the issue had long been more or less resolved. Even in the early testimonials of men such as Polycarp we can see the early church's attitude and reverence regarding the Holy Writings. Of course as the episcopacy developed, with its seeds apparent even in Ignatius of Antioch, and as battle was engaged with the Gnostics, the emphasis shifted from Scripture to Episcopal Guardianship. If Sola Scriptura is incorrect, then I will grant a Charismatically Inspired Episcopacy is the only other valid option. Otherwise on a practical level, tradition, the commandments of men will and did take over and ultimately overthrow the gospel. More on the Sufficiency and Insufficiency of Scripture later…
So what's the goal? Do we want to reform the church to the 1st century? Can we? Should we even want to do it? Again, I can accept a certain logic to the necessity of organic development, expansion etc… BUT if that's so, then we must abandon Sola Scriptura…….the medieval dissenters were wrong, but then so was The Reformation. If development is legitimate as a source of authority, meaning the source of authority expands, or revelation is improved on as the old systematicians used to put it, then the Reformation doesn't really have a leg to stand on.
With Constantine a macro-shift took place in the life in the church leading to a unity, an often dynamic unity between church and state. This necessitates a redefinition of the Kingdom of God as given by Scripture. The Kingdom which was once not of this world (which cannot just be restricted to Johannine categories of lust of the eyes, flesh, and the pride of life), now was the world transformed. Or to put it another way the city of man was transformed into the city of God. If this was not legitimate, then it was the beginning of apostasy, which is exactly what the medieval dissenters asserted.
This will be a recurring and expanding discussion. For starters if the concept is new, a resource as simple as Wikipedia will help. There is a helpful article on Constantinian Shift. This and other concepts are also a major theme in Verduin's "The Reformers and Their Stepchildren." There he discusses another key concept, Sacralism. The book is not above criticism and I would take a definite exception to a few points he makes, but the book has largely been left alone. Its premise and conclusions are titanic. It is perhaps one of the most original (in a good sense) and iconoclastic works of the last century in the Protestant world. F.N. Lee wrote a scathing critique of it which I may interact with at a later date, but those familiar with his work will understand my lack of regard and consideration.