Hoooo boy. That's a big question and though I will attempt to answer it as thoroughly as my schedule allows, I will inevitably have to make some generalizations here. Anyway, here goes.

It's impossible to pinpoint a single date, but there are a few key events here that largely took place in the fourth century that could be considered to be major turning points for Christianity in the Roman Empire, although Christians were a significant group in certain places in the Empire even before the fourth century. Early on, Christianity was either mostly ignored by the Roman authorities or actively persecuted. The reasons for this are many, and I will address some of the bigger ones:

Among other things, Christianity was problematic because being strictly monotheistic they refused to make their religion "yet another cult" within the religions of the Roman Empire (the Romans were pretty much okay with adopting all kinds of foreign gods, for example the Egyptian goddess Isis was fairly popular in Rome) and they usually refused to partake in any kind of traditional cult acts (such as offerings) that were expected of Roman citizens regardless of what they believed.
The Romans believed that in order to maintain the pax deorum (or 'peace of the gods' - the peaceful relations between mankind and the deities), the proper religious practices should be observed. Note that it wasn't so much belief that was required here but rather compliance with the accepted Roman tradition, as there are examples of Christians who didn't have qualms with sacrificing to pagan gods and thus saved their lives.
This negative stance towards the common cults in the Empire in early Christianity (which had a lot in common with the so-called mystery cults of the time) also made the religion appear to be sinister and dangerous in the eyes of many common pagans in the Empire.
Some emperors also used the Christians as an easy scapegoat, for example Nero shifted the blame for the Fire of Rome in 64 to the Christians after rumours started circulating that Nero himself started it.

Then in 311 and 313, two Edicts were issued by the emperors Galerius and Constatine, respectively. The most important of these is the Edict of Milan in 313, which essentially 'legalized' Christianity after centuries of persecutions. From Constantine's conversion and onwards, the imperial policy towards Christianity would shift dramatically from persecution towards active promotion, and so it soon became an attractive option for people to convert to Christianity (at the time of the edict the Christians were still a minority in the Empire). During the fourth century, Christianity shifted from being a persecuted minority religion to an integral part of the Roman (and thus, 'civilized') world.

It became even more attractive when during the reign of Theodosius I (379-395) a series of decrees were issued that effectively banned all non-Christian religious practices. From this point, paganism would rapidly decline within the Empire, though at that point the Empire itself was already experiencing a strong decline and the various 'barbarian' peoples settling in the Empire were usually pagan (or Arian Christian, which was considered a heresy), and they maintained their beliefs until their conversion at various points during the early medieval period, after the fall of the Western Empire.

In the Eastern Empire, Christianity would firmly take root and endure, evolving into what we now know as Greek Orthodox Christianity, and it remained the state religion there until the conquest of Constantinople in 1453 by the Ottomans which finished off the Byzantine (Eastern Roman) empire once and for all.

Sources:

An Introduction to the Ancient World by L. de Blois and R.J. van der Spek
Medieval Europe: A Short History by Judith M. Bennett (11th ed.)
Why were early Christians persecuted? by G.E.M. de Ste Croix in: Past & Present 26 (1963)

Also from my undergrad classes in Ancient and Medieval History at the Uni of Amsterdam.