We know very little about Minucius Felix. He was a lawyer, almost certainly from North Africa although practicing in Rome, who converted to Christianity; he lived in the late second century or early third century (he is mentioned by Lactantius and seems to be alluded to by Cyprian, so he is no later than the middle of the third century). The Octavius is his only known work, although Jerome seems to say that he also had another work, Of Fate, which is not extant and is mentioned by no one else. That's about all the reasonably direct information we have.
The Octavius only survives in one manuscript tradition (based on a ninth-century manuscript in Paris); it was overlooked for a long time because it is bundled with Arnobius's Against the Nations and was only discovered not to be part of that work in the sixteenth century. It has a number of obvious links to Tertullian's Apologeticus, so there would be room for extrapolating more if it can be determined which of the two has priority; that would, for instance, give us a better sense of dates for the work. If Octavius is prior to Apologeticus, this would put it prior to the third century, if it is posterior, this would put it after the second. The scholarship has gone back and forth on this. It was originally thought that Octavius was influenced by Apologeticus; then beginning in the seventeenth century, the priority of the Octavius seems to have become dominant, in part because Minucius's handling of classical authorities is more precise than Tertullian's and because Minucius seems to be describing, even if we assume he is embellishing, a real event. No arguments on either side are really definitive.
Besides (possibly) Tertullian, Minucius is quite clearly heavily influenced by Cicero, in structure and in style as well as in some of his more general themes.
You can read the Octavius at Tertullian.org or Early Christian Writings. C. Francis Higgins has a nice article on Minucius Felix and the Octavius at the IEP. Paul Lejay also has a nice, if now somewhat dated, article in the 1911 Catholic Encyclopedia.
The dialogue is a discussion involving three friends:
Marcus Minucius Felix: The narrator, he acts as the 'judge' in the debate between Caecilius and Octavius. His name is on monuments in Tebessa and Carthage, although it's unclear whether these refer to the same Minucius. At the beginning of the dialogue, he is practicing law in Rome.
Caecilius Natalis: A pagan. There is a monument with his name on it at Cirta, in North Africa, where the dialogue claims he is from; whether this is the dialogue's Caecilius or a family member, the Caecilius mentioned on the monuments was an active participant in the pagan life of Cirta.
Octavius Januarius: A Christian. His name is on a monument in Saldae, although, as with the other two, it's unknown whether this is the same Octavius or simply related to him.
The dialogue is nostalgic in tone (influenced by Cicero's De Oratore). Minucius, reflecting on his friendship with Octavius, who has died, is brought back to the time when Octavius converted their friend Caecilius to Christianity. Octavius came to visit Minucius in Rome, and as the courts were on hiatus, they went down to Ostia for bathing and relaxation. They are walking to the beach with Caecilius, when Caecilius sees an idol of Serapis, and salutes it with a kiss. Octavius makes a mocking comment on the gesture, scolding Minucius for letting his friend remain in ignorant superstition. They spend some time enjoying themselves at the beach, with Octavius telling various stories about the sea. As they are returning along the shore, they come to the docks and see the boys skipping shells on the waves, each one trying to outdo the others. At this point Minucius notices that Caecilius is a bit too quiet, and asks him what is wrong. Caecilius says he is bothered by Octavius's comment, less atbeing called ignorant than at the the fact that Minucius had been rebuked for it, and he proposes, if Octavius is willing, that they sit down and argue it out. (There is undoubtedly a narrative parallel between the boys and the men here. The men are adult lawyers, so their form of skipping stones in friendly competition is to argue.) Minucius will be the judge; Minucius is a Christian convert, but Caecilius asks him to be as completely impartial as he can. Caecilius gives his arguments against Christianity; he's put in a much better mood by it, likely by the opportunity to show off his oratorical skills. Minucius scolds him for thinking so highly of himself when Octavius hasn't even spoken yet; Caecilius rebukes him lightly for not maintaining his impartiality, but Minucius responds that they should be concerned less with the eloquence than with the truth of the discourses. Then Octavius gives his defense of Christianity. After he finishes, they are silent a bit. Minucius is impressed by Octavius's ability to express what is easier to feel than express. Caecilius admits himself outmatched, but declares himself a victor of the sort, in that Octavius has led him to understand the matter better. As the sun is setting, Caecilius proposes that they discuss the matter more deeply the next day, and they head home.
The story that frames the debate is an extraordinary literary capturing of the kind of nostalgia that is associated with close friendship, and in and of itself gives the dialogue a charm and interest that accentuates the actual debate.
The apologetics of the work is primarily sociological, not theological. Certain broader philosophical issues, particularly knowledge of the goods and the existence and nature of providence, play a significant role (and connect the dialogue to its Ciceronian forebears), but for the purposes of the argument, they are primarily there to assist in determining whether Christian life is reasonable.
Caecilius argues a version of the Academic position that we only know the probable, not the true, due to the uncertainty of human life, and criticizes Christians (often uneducated) for being overly dogmatic on matters beyond human ken. Either human life is governed by chance or, if it is governed by nature, we do not have certain knowledge of it; because of this, the safest course is to stick with the customs of your forefathers. This is all the more the case in that the Romans had risen to greatness in preserving the religion of their fathers, and failures to respect the auspices had often led to disaster.
Thus the presumption should be for the pagan religion, which has the consent of nations. Christians, however, reject this and do so not by argument but by conspiracy. And in their secret societies, they are rumored to do horrible things: worship the head of ass, engage in incest, worshiping the genitals of their priests, having orgies, drinking the blood and eating the flesh of babies, and (even worse) worshiping a criminal and the cross on which he was executed. If such rumors are not true, why do Christians go around in such secrecy? What is more, their God is a strange God nobody has ever heard of before, yet they think He is a busybody who is nosing about even in peoples thoughts, all over the earth, and they threaten people by saying that He is destroying the world, which everyone knows to be sempiternal, with fire. On top of that, they claim that bodies will be resurrected after the destruction, and promise themselves blessed perpetual life after death while condemning everyone else to everlasting punishment, not because of what they deserve but because of what the wicked judge they pretend to be God decides beforehand. Nor does their account of resurrection even make sense -- the body was already destroyed, so it can't be resurrected, and if it's a new body, then you are not same person.
All of this seems clearly to be made up so as to help them veil to themselves what seems to be the real situation: they are often wretched, destitute, and homeless -- apparently with their God not much caring -- whereas the supposedly inferior pagan Romans live lives of enjoyment and happiness in comparison. Christians are thus acting neither modestly nor wisely, nor with proper regard for the limitations of the human mind. Christians would be better off imitating Socrates, who refused to speculate about divine matters, and who showed (confirmed by the Oracle at Delphi) that confession of one's own ignorance is wisdom -- the foundation of the Academic approach to philosophy.
Octavius responds by saying that Caecilius's argument is inconsistent and erratic. Parts of his arguments depend on a deep skepticism about divine things and parts only have real force against Christians if we know divine things fairly well. He can, however, respond to all of it from a single and consistent position. Caecilius complains about the Christian lack of education and wealth, but fails to grasp one important thing: "all men are begotten alike, with a capacity and ability of reasoning and feeling, without preference of age, sex, or dignity." The capacity to be wise is not a matter that is hostage to fortune, and you do not become wise by being rich, nor even by mere study. Though poor and uneducated, they may still be wise. Their position should be judged not on their circumstances but on its truth.
Caecilius is right that we should know ourselves and be aware of our limitations, but he draws the wrong conclusions from this, because he should instead recognize that things in this world are so interconnected that we need to know divinity in order to know humanity. It is absurd to exaggerate the extent to which the world is governed by chance; we see quite clearly that the world is very orderly, in a way that suggests divine reason. There is a divine providence. What is more, this is not just a generic providence but one that extends to the parts of the world, because you still see the orderliness when you focus on the details. Our real limitation is that the God who must exist to do this is too much to know. We should not, like the pagans, diversify this by many names; we should simply recognize that any names we give will be inadequate. And this, Octavius says, is the common consent of nations: people spontaneously address God just as God. "And they who speak of Jupiter as the chief, are mistaken in the name indeed, but they are in agreement about the unity of the power." The poets and the Stoic philosophers, too, often talk in this way, and many of them agree with the Christians that this divine power pervades all. What is more, Plato's doctrine in the Timaeus is very similar to what Christians say of God. So in fact there is plenty of philosophical authority in support of something broadly like the Christian position, and we should take this to be more seriously than just the traditional opinion, because know that people have often fallen easily into false beliefs. And if we're talking about absurdities and inconsistencies, it's not as if you can't come up with a long list of traditional pagan beliefs that propose bizarre things or are connected to strange or inconsistent stories. And the real shame is worshiping idols, stone or wood, that are made by human hands, and making them the center of pagan rites.
Caecilius had attributed the greatness of the Romans and the value of their auspices to their religion. Octavius replies that the Romans have a long history of impiety, and a simpler explanation is that they were so successful because they were violent and terrifying. Likewise, Caecilius's appeal to auspices is highly selective; there are many cases of people taking auspices and meeting a bad end. This is not to say that the argument has no force: auspices do sometimes seem to have touched on the truth, impressively even. But this is because there are spirits who know more than we do and meddle with human matters, like Socrates's numen. If there are angels and demons, then the occasional impressive successes of the auspices can be explained; but these spirits are everywhere recognized as intermediaries, not gods. And these demons are themselves what stirs up the prejudice against Christians, so that people hate them before they even know them.
This brings us then to the claims that Christians engage in horrid crimes in secret. Octavius notes that he himself (and perhaps he's including Minucius, as well) had once believed such things. Even as he would defend people from charges of crimes like these in court, he did not even bother to think that Christians might have something to say in their defense. Many of them are not even very plausible, like the oncephalic calumny, that Christians secretly worship the head of an ass, but even if it were true is not so very from Egyptian paganism. Christians do not worship a criminal; they believe the man on the cross was unjustly killed, and, moreover, was God; but, again, pagans cannot attack Christians for these things without answering for the fact that their are pagans who do analogous things. Similar things can be said of drinking the blood of infants: this is not right, but even if it were true, pagans have a long history of exposure of infants and of abortion. Similar things go for the charge of incest;Christians have strict standards in favor of marriage, and call each other brother and sister not out of incest but out of recognition of God their Father, and out of mutual love for each other. In every case, the pagans charge is that Christians do things, leaving no evidence, when there is plenty of evidence that pagans do things analogous to what they are attacking. Nor is it even true that Christians are engaged in secret conspiracy. They don't blabber about everything, to be sure, but they don't think God can be confined to a building, and they think the proper sacrifices to God are not showy animal sacrifices but virtue, innocence, justice.
As to the conflagration and the resurrection, the idea that the world will perish in fire is entirely consistent with the common recognition that all things that begin also perish, and the Stoics too hold that this world-order will perish in fire. Nor is it absurd to think that man, who has been formed by God, can be re-formed by Him. We are still in the winter of the body; there will be a spring. And it is not difficult to find pagan poets and philosophers peaking of reward and punishment in the afterlife, as well. To ignore God is sufficient for punishment, to know Him is sufficient for pardon; but even if that weren't so, Christian standards of morality are higher than pagan standards. The reward and punishment is not a matter of fate; men are free; it is our doing that is judged, not our being. And the adversities of this life increase us in virtue, especially martyrdom. Christians set aside the pagan customs not out of fear and a desperate attempt to comfort themselves, but for something better. If the Academic philosophers think otherwise, let them be pitied for it; Christians have obtained the truth that the Academics sought badly.
Caecilius concedes the argument; Octavius has given a reasonable answer to all of his objections. He accepts that it is reasonable to believe in providence and that the Christian way of life is sincere.
* Part of the dialogue is an argument over the nature of providence, with Caecilius suggesting as possible an Epicurean view and Octavius, without getting too technical, drawing on Stoic arguments for divine providence. Octavius's argument seems heavily influenced by Cicero's presentation of the Stoic argument in De natura deorum.
* All of the anti-Christian calumnies Caecilius notes are calumnies for which we find independent evidence elsewhere, sometimes in slightly variant forms. Some of them, like the onocephalic calumny, were Roman calumnies against the Jews that the Christians inherited. Others, like the cannibal feasts, the incestuous orgies, and the worshiping of a criminal are obviously garbled versions of Christian doctrines.
* One of the things that is important to the dialogue, and emphasized by Minucius's intervention between Caecilius's and Octavius's speeches, is that friendship competes for truth (veritas) and not praise (laus). Caecilius, despite his immediate objection, affirms much the same by the end of the dialogue: they are not in a competition with a mutually exclusive victory, but Octavius can conquer by showing forth the truth, yet Caecilius can still be a victor by coming to see it. The only victory that matters when friends are involved is truth.