Paul writes to Philemon about the slave of the latter, whose name was Onesimus. Paul had come across Onesimus during his time in prison, and in that period Onesimus had become a Christian. Now Onesimus seems to have been separated from his master Philemon, so Paul writes a letter to Philemon about how he will now return Onesimus to him. Yet he returns him to Philemon with a request, namely that he willingly allow Onesimus to return to Paul to help him (13).
Of course, this presupposes that Philemon will accept Onesimus upon his return. In the ancient world, runaway slaves were not always treated very kindly. Sometimes they would have the letters F U G branded on their forehead; these letters came from the Latin word for fleeing, fugare, which indicated that they were runaways. A runaway slave was hardly a very honorable or respectable person, even if his flight was justified and reasonable. Some of them would even join gangs, bandits, pirates, etc., because they would not fit in among polite society. On the other hand, some masters would even kill their slaves upon return if they had fled.
Paul calls upon Philemon to accept Onesimus kindly. Yet it seems he does even more than that; arguably, he wants Philemon to accept Onesimus, and furthermore to give him freedom, so that he can serve Paul. He writes:
Perhaps this is the reason he was separated from you for a while, so that you might have him back forever, no longer as a slave but more than a slave, a beloved brother (15-6).
Now it is controversial whether Paul is asking Philemon to give Onesimus his freedom or not. But to my mind, that is the most likely explanation. He explicitly says that Philemon might have Onesimus back "no longer as a slave." If he is no longer a slave, then he is a freedman!
Some persons think this goes against Paul's words to slaves and masters in his other epistles. For instance, in Ephesians and Colossians he tells slaves to obey their masters, and masters to treat their slaves well. There it would seem that Paul does not want to undo the slave-master class system, but only to teach the two parties how they ought to behave in light of their salvation in Christ. But that is too easy an explanation, since in many ways what Paul tells the slaves and masters does undo the slave-master system. If masters are to think of slaves as equal to themselves, since they too are slaves of Christ and God (Eph 6.9), then they are no longer masters over their slaves. If they are not given freedom to do as they please with their slaves but must see them rather as brothers and sisters, as fellow members of the body of Christ, etc., then they can't think of them any longer as slaves -- as mere human property.
What Paul does in those letters is plant seeds for the undermining of the system of slavery altogether. He does not explicitly demand that every slave be freed, but what he does command is crucial for guiding the growth in understanding of his Christian audience towards the realization that slavery is anti-Christ. But in Philemon he arguably does command Philemon to free Onesimus. Perhaps he does this here and not elsewhere because he knows Philemon more personally.
Someone might want to object that Paul does not demand Onesimus' freedom so much as his acceptance and forgiveness as a Christian, while remaining a slave. They might say that Paul wants Philemon to accept him as "more than a slave," which allows him to remain a slave while being treated more honorably than expected. But the phrase "more than" is not always inclusive: if I am more than a boy, but a man, then I am not a boy at all; if I am more than an ordinary worker, but a CEO, then I am not an ordinary worker at all; if I am more than a slave, but a freedman, then I am not a slave at all. Sometimes one thing is more than another while not including it. And for me it is rather obvious that Paul's language -- no longer as a slave -- suggests he means to use the phrase "more than" in an exclusive manner.