Contrary to popular belief, the Romans were normally quite tolerant of other religions, and would have been towards Christians were it not for some of the practices of Christians themselves. Christianity was actually considered something of a superstition rather than a true religion. Early Christians held firmly to the idea that all gods other than theirs were false gods and should not be worshipped. This was at at time when the Emperor had achieved god like status and was often worshipped as one. By refusing to pay proper homage to the Emperor, or the official Roman Gods, the Christians were considered guilty of treason. The common belief in Rome at the time was that proper homage to the Gods prevented bad things from happening, such as losing a battle. If one did not pay proper homage, then those bad things would indeed happen. By refusing to worship the official Gods, including the Emperor, the common belief was that Christians were causing bad luck. Christians also spoke of a "Kingdom not of this world" which was worrisome for Roman authorities. The fact that the sect continued to grow despite official disapproval was also troubling. During the great fire of Rome, Nero blamed the fire on the Christians, since they were presumably "anti-Emperor," and under any circumstances, had brought bad luck by refusing to worship the Roman gods.
Although Constantine legalized Christianity by the Edict of Milan and professed himself to be a Christian on his deathbed, there is substantial argument that he did so for political reasons only. Christianity continued to grow in popularity even though they were often persecuted; and his decision was for pragmatic reasons.
There are two major reasons why Christianity was persecuted in the Roman Empire. (You should, however, be aware that the persecution was not constant and it was not the same in all parts of the Empire.)
First, Christianity was a monotheistic religion. Christians were fairly militant about not doing anything that would even imply that there were any other gods than God. This meant that they would not participate in Roman religious festivals and observations that were believed to bring good fortune to the Empire. This made them, in a sense, subversive because they refused to help protect the Empire in this way.
Second, Christianity was a new religion. The Romans generally tolerated old faiths, but they saw Christianity as a new thing and therefore as a mere superstition. They did not accord it the respect they gave to older religions.
These were two major reasons why Christianity was, at some times and places, persecuted in the Roman Empire.