Different times were for different reasons.

For example, Nero persecuted Christians because he needed a scapegoat for the great fire of Rome. This was a relatively limited persecution though.

A significant period of persecution was under Decius. This was done because the Empire was in a time of crisis. Decius felt he needed to reassert to his citizens what it meant to be Roman, reestablish a 'Roman' identity and bring Rome back into the favour of the gods. His persecution was based around forced conversion/denial of Christianity, and torture or death for those that refused to try and reassert traditional conservative Roman values in am effort to bring back stability to his society.

Diocletian also persecuted Christians a while later for similar reasons to Decius.

It is important to note there was no specific long running systematic war against Christians in the way many seem to think. There were certainly long periods of systemic persecution, but there were also long periods of peaceful coexistance and tolerance. Essentially, Christians were a small but growing sect, and from time to time it served an Emperors purpose to either use them as a scapegoat, or cast them as an 'enemy of Roman values' to help strengthen the notion of Roman values. After all, its much easier to reaffirm Roman values by persecuting someone rather than encouraging people to just 'be more Roman'.

Though the specifics vary, the reason really did not: Christians were disruptive to the social order. So were the Jews. Both were persecuted.

Rome--and this is a simplification, but broadly true--a easygoing relationship with the religions of the people it assimilated. For example, the Romans rather enthusiastically imported the Egyptian goddess Isis into their own pantheon--a pantheon that was itself based on Greek tradition. Almost everyone was allowed to maintain their existing gods after they joined the Empire.

The main stipulation the Romans laid out for their client states was that the Emperor was to be treated as a god among whatever other gods might be hanging around. For polytheistic societies (that is, almost all societies at the time) this wasn't a big deal. What was one more god among many?

The Christians and the Jews, however, only acknowledged a single God. This was the basic tenant of their beliefs. To allow Caesar into the temple would be a terrible sacrilege.

For the Jews, with their established state, this meant open (and frequent) rebellion. For the Christians, who had no state--or particular cultural identity--this meant that they were forced to create a sort of secret society. A secret society that focused on recruiting from slaves, the poor and disposed, initially.

From the Roman perspective, the Jews were clannish, stubborn and inscrutable while the Christians were a bunch of agitators and malcontents (remember, the Roman elite lived in terror of their slaves and/or underclasses revolting). As both groups were living inside the Empire and the Empire was comfortable with casual violence in ways modern societies are not, then you can see how they'd be popular targets for persecution.

A large part of it was that Christians often refused to worship the Emperor. Since the Emperor was a living god to Roman citizens, this was seen as treasonous. They also did not participate in public religious festivals, and many Romans misunderstood the Eucharist (through anti-Christian propaganda), seeing it as actual cannibalism.

Kenneth Scott Latourette, A History of Christianity, p82.