I'm pretty sure that it all started with Constantine the Great, the Emperor of Rome. He was the first Christian emperor of Rome and it was because of him that Rome was unified and all other would-be emperors were dispatched.
Just before a great battle against Maxentius, a wanna-be emperor, Constantine was visited by Christ in a dream. He painted Christ's symbol on the shields of all of his men, which were four times less than that of Maxentius', before battle. Constantine and his men defeated Maxentius' men and, as they were fleeing over a bridge, it collapsed and Maxentius and all of his thousands of troops were drowned. Constantine believed that his victory was entirely attributed to Christ and so named himself the Emperor of the Christian people.
For some time after this Christians were rarely persecuted. Emperor Constantine still allowed Pagans their gods especially the sun god whose face or name, I believe, is on one of his coins. Property was returned to Christians and they were, once again, allowed to purchase land and worship openly.
As time went on Constantine became more involved with the Christian church. He sought to resolve theological disputes among the church.
In this role he summoned the bishops of the western provinces to Arelate after the
Donatist schism had split the church in Africa. He worked to solve debates peaceably but brutal enforcement of the decisions reached at such meetings was not uncommon. Following the decision of the council of bishops at Arelate, donatist churches were confiscated and the followers of this branch of Christianity were brutally repressed. Evidently Constantine was also capable of persecuting Christians, if they were deemed to be the 'wrong type of Christians'.
Licinius, Constantine's brother-in-law and emperor in the east, I believe, began to oppose him and persecute Christians again and even started ejecting them from their positions in the government. The two fought repeatedly. Finally Constantine defeated and later executed Licinius, becoming the sole ruler of Rome. Soon afterwards Pagan sacrifices were outlawed and money taken from Pagan temples was given to Christians to build churches. Gladiator sports were outlawed and Jews were no longer allowed to own Christian slaves.
He made some pretty severe laws and had pretty harsh taxes, but the people loved him, mostly because of the prosperity and peace he brought to Rome.
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And thus ends the debate
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Christianity made it easy for anyone to join, and it promised so much.
Also, many religions required an initiation; They were costly and exclusive "clubs".
Christianity just required baptism, and it was open to the lowliest of the low.
We touched on this idea briefly in my first semester of Anthropology: Christianity was created for the poor and Catholocism was a religion for the rich. Constantly in the Bible you read about how very few rich people will make it into Heaven. It was easy for people to believe in a god that was protecting them and providing for them when they had nothing. They were able to take comfort in the idea that in the end "the last [would] be first and the first [would] be last". God even say that "it is easier for a camel to go through the eye of a needle than for a rich man to enter the kigdom of God".
While Catholics have lavish and expensive temples and robes and ceremonies, it would seem that Christians pride themselves on their simplicity, "come as you are".
Perhaps because the poor outnumbered the wealthy it was easy for Christianity to catch on after Constantine made it okay to practice openly.