Explanation: While later Christian apologetics often focus heavily on a narrative of orthodoxy-mandated imperial Roman persecution, the reality of Roman persecution of Christians is much more complex. The Romans, generally, were not overly concerned with what people believed, nor what religious practices they engaged in in most cases.
The Christians met an unfortunate junction of several circumstances which led to intermittent localized persecutions.
First, that they were almost always on bad terms with the local Jewish community. The Romans generally preferred to let peoples take care of their own affairs, and this went double for the Jewish community, whom the Romans did not always fully understand the nuances of. This meant that the normal exemptions available to the monotheism of the Jews were not available to the early Christians - as the local Jewish community was generally quick to repudiate them as weird foreign cultists, not Jews. Interestingly, there is some speculation (though far from a settled conclusion) that the option of paying the Jewish Tax - what the Jews paid the Roman Empire in lieu of making sacrifices - was available to Christians, but Christians did not avail themselves of it, possibly because they insistently did not see themselves as Jews, and thus could not countenance declaring themselves as such to pay the tax.
Second, that their secretive nature - primarily to avoid local prejudices, especially from the Jewish communities that they took many of their early converts from - rumors spread very easily about their practices. One of the earliest Roman accounts of Christians, by Pliny the Younger, records rumors that they engaged in cannibalism and incest - most likely due to the metaphor of the eucharist ('consuming the body and blood of Jesus Christ') and the tendency of them to refer to one another as 'brother' and 'sister'. To Pliny's credit, he determines that these rumors are baseless, and all that the faith is is some obscure, seemingly harmless superstition, and wrote to the Emperor to ask for further instructions. These rumors would not die with this early disproval, however, and lasted all the way until Christianity took over the Empire entirely.
Third, that they regarded the normal rituals of the Roman state to be pagan and unlawful for them to engage in. This is not something the Romans encountered often - most ancient faiths had no issue performing rituals to other gods, so long as it didn't violate any other taboos. To the Romans, this was the equivalent of refusing to pledge loyalty to the Roman state - if you will not make your loyalty oaths by the gods, clearly you are planning on breaking or ignoring the oath!
Fourth, that in most cases, Christians were not Roman citizens in the early years of Christianity. This meant that their rights were the rights of provincials - and thus very... limited. 'Obstinacy' was a common charge leveled against Christians in this early period - which sums up to "A filthy provincial refusing to listen to a Roman official." An executable offense!
And fifth, that like most minorities, they were easy scapegoats for local discontent. Mobs and governors alike found them convenient to blame local problems on, whether natural or social.
The incidents of actual imperial persecution - in which some manner of Empire-wide censure was placed on Christians - are actually very few, and all of them not long before Christianity took over the Empire itself.
The first being under Decius in 251 AD - possibly accidentally. Emperor Decius demanded that all the subjects of the Empire sacrifice, in the name of the Emperor, for the health of the Empire. This was a means of reinforcing loyalties, as Decius had only recently usurped the position of Emperor. Unfortunately, Christians did not particularly care for sacrificing in the name of the Emperor, regarding it as a pagan practice. Many refused to perform it, and were subject to punishments. Whether Decius intended this is unknown, as he died in battle very shortly afterwards.
The second being under Valerian in 257 AD. This one was undoubtedly intentional - demanding that Christian clergy and any high-ranking Christians make sacrifices to the gods to prove their loyalty, or else be executed.
The third, and most severe, being under Diocletian. It stretched a number of years and disadvantages, but the worst period being from 303 AD to 306 AD, wherein several thousand Christians were executed for no other reason than their faith. After this final persecution, the Emperor Constantine I came to power, and made Christianity the religion of the Empire - against the will of the majority of its inhabitants.