The impact of the Church's triumph (c.300-500)

The city is full of mechanics and slaves, who are all of them profound theologians and preach in the shops and in the streets.  If you desire a man to change a piece of silver, he informs you wherein the Son differs from the Father; if you ask the price of a loaf, you are told by way of reply that the Son is inferior to the Father; if you inquire whether the bath is ready, the answer is that the Son was made our of nothing. — Gregory of Nyssa

The favor Constantine and his successors showed the Christian Church increasingly made it the state religion of the empire until 393, when the emperor Theodosius ordered public worship in the pagan temples to be ended throughout the empire.  Christianity had triumphed, but success would also bring its problems.
The root of the Church's problems lay largely in the heavy persecutions of the third century that did two things.  For one thing, they created a more decentralized Church by driving into hiding Christians who had lost contact with one another.  On the other hand, the persecutions also helped lead to the triumph of Christianity as the virtual state religion by giving it publicity that attracted converts.  In addition, as Christianity gained popularity, formerly pagan intellectuals joined the Church in greater numbers and started grafting pagan, especially Greek, philosophies onto Christianity.  These factors would contribute to two very different lines of development in the history of the Church: the spread of religious disputes and heresies and the rise of monasteries.

Religious disputes and heresies

One of the more confusing aspects of Christianity was the nature of the Trinity of Father, Son, and Holy Spirit and the relation of the three parts to one another. During the persecutions, communities of Christians isolated from one another had developed different ideas on the Trinity.  Christianity's legalization meant these different congregations which persecution had forced underground (literally in some cases) now could come out into the open to find they had very different ideas on this point.  Added to this was the growing number of intellectuals joining the Church who, instead of just accepting Christianity as a simple religion, saw various subtle interpretations of the concept of a triune god.  Confounding the confusion was the vague wording of the Bible itself, which also led to different points of view.
The most serious of these disputes centered on the relationship of the divine and human natures of Christ.  The first of these, the Arian dispute, flared up soon after Constantine had legalized Christianity.  The issue was whether Christ, being the begotten son of God the Father, was co-eternal with the Father, and thus fully divine.  An Egyptian priest, Arius, said he was not co-eternal with the Father.  The Arian view, as it was called, spread widely throughout the Roman world, causing heated arguments and even violence.  Therefore, instead of creating a unifying factor for his empire by legalizing and favoring Christianity, Constantine had unleashed a wholly new type of controversy that threatened to tear the empire apart.  Given the Church's close relationship now with the Church, Constantine and later emperors felt they could not tolerate religious disputes and heresies.
There was a general and unfortunate pattern to these religious disputes that made a correct solution to them practically impossible to achieve.  A new interpretation of Christianity would pop up and gain converts.  This would lead to arguments and at times bloodshed.  A church council, backed by the emperor, would denounce the new belief as a heresy (wrong belief) and either exile the heretics or persecute them within Rome's borders in order to preserve the public peace.  Unfortunately, dealing with heresies in this manner usually backfired much as imperial persecution of Christianity had backfired a century earlier.
Today, many people may wonder why people and governments got so emotionally involved in these disputes.  The answer revolves largely around Christianity's exclusiveness.  It was seen as the only true religion and path to salvation.  Along those same lines, one had to have exactly the right belief in order to be saved.  Just the slightest deviation from that belief could mean eternal agony in Hell.  The Roman government also believed in supporting the exact right belief in order to ensure God's favor and protection.  Tolerating heresies could lead to God's disfavor, and any military defeats or natural disasters were interpreted in that light.  Also, since the Roman Empire had tied its fortunes securely to the Christian Church, its religious and political policies were tightly interwoven.  Tolerating religious heresies was seen as the same as tolerating treason.  Therefore, from the later Roman Empire to the early modern era (c.300-1700), religion and politics went hand in hand, and a decision in one realm generally had serious implications in the other realm as well.  The history of two of these heresies, the Arians and Monophysites, especially shows this mentality & its results in action.
In the case of Arius, Constantine called a council of Christian bishops together at Nicaea in Asia Minor in 325.  Arius was logically shown to be wrong, his beliefs were declared a heresy, and he himself was exiled.  Arius then went to the northern tribes whom he converted to his brand of Christianity.  A century later, when these tribes conquered the Western Roman Empire, they did it as Arian Christians.  Now it was the Catholic Christians, who made up most of the Roman population, which were often persecuted.
Another heresy, that of the Monophysites, was suppressed in Egypt, Syria, and Palestine, which led to strong undercurrents of resentment and even rebelliousness against the Roman government.  When the more tolerant Arab Muslims invaded these provinces in the 600's, instead of meeting stiff Christian resistance, they found the populace oftentimes welcoming them against Roman oppression.

Poverty, chastity, and obedience: the rise of monasteries

The success and favored status of Christianity also brought other problems.  When Christianity was an outlawed religion, the motives and sincerity of its members were rarely in doubt since there was nothing to gain and plenty to lose by joining the Church.  When Christianity became the favored religion of the Roman Empire, all that changed.  There was an influx of new members joining for reasons of social, political, or material advancement.  Also, the influx of intellectuals who grafted pagan philosophies upon the Christian faith was complicating the religion almost beyond recognition.  The purity of the Church's membership was becoming seriously diluted.
This upset many of the more devout members of the Church, and they wanted to purge it of such worldliness.  Since they could not drive the new members from the Church, they retreated into the desert to live pure Christian lives away from worldly temptations.  In order to cleanse themselves of their sins, some of these men performed incredible feats of endurance nearly to the point of self-destruction.  One such feat was to sit on the top of a pillar for years at a time.  Another was abstinence from food almost to the point of starvation. As word of these "super-hermits" spread, other devout Christians moved out to the desert to be near them and share in their holiness.  Soon the desert was so crowded with these people that they had to be organized into communities called monasteries.  In the East, St. Basil was the man who established the first monastic rule.
In the West, it was St. Benedict. After a fairly sinful and dissolute youth, this man launched a career of violently trying to purge himself of his sins.  At last, he arrived at a more moderate concept of Christianity and formed a monastic order known as the Benedictine Rule.  The Benedictine Rule reflected its founder's more moderate views, though it was still strict by modern standards.  A new monk took three vows: poverty (no material possessions), chastity (clean living), and obedience (to God and the superiors in the monastery).  The day was divided into roughly equal parts of prayer, work, and rest.  Incredible acts of self-torture or self-denial were not expected.  Instead the monk worked around the monastery and in the fields, the belief being that idle hands are the devil's playground.  Our own modern work ethic is directly descended from this idea.
The moderate expectations of the Benedictine Rule led to the spread of their monasteries all over Western Europe.  As the orderliness of the Roman Empire gave way to the anarchy of barbarian rule, monasteries and monks would provide the one shining light of civilization in the West.  These quiet and vigilant men bravely spread the word of their religion beyond the frontiers of the old Roman Empire, thus spreading civilization to new areas as well as preserving it in old ones.  Monasteries were also the main centers for any kind of social and economic relief in the Dark Ages.  The poor and destitute looked to them for food, shelter, and protection.  The sick looked to them for hospital care.  And travelers looked to them for safe havens on their journeys.
Another important and somewhat ironic aspect of monasteries was that many of the pagan intellectuals whom the hermits had originally tried to avoid were now showing up in the monasteries in an effort to flee the growing anarchy as the Roman Empire fell.  These men, who had received a pagan classical training brought their love of that pagan culture with them and devoted much of their time to copying pagan works of literature.  Thus ironically, monasteries, which started as a somewhat anti-intellectual movement, were the primary agents for preserving ancient pagan culture during the Middle Ages by carrying on this tradition of book copying.
Throughout the Middle Ages, the contributions and monasteries and of the Church overall were immensely important to our culture.  The early Church was very much a part of Roman Civilization and absorbed a good deal of it into its own theology and ritual as shown by keeping the mass in Latin until very recently.  As the Roman Empire faded from history, the Christian Church survived to carry on the Roman heritage along with its own unique contributions to Western Civilization.