“The Apostolic Fathers” is a fascinating collection of early Christian writings. How early? Nobody really knows, with the enthusiasts placing some of the texts in the latter half of the first century. Others weren't written until the second century. There have also been some debate over which of the texts are genuine.
What makes “The Apostolic Fathers” important is that they document the formation of Church tradition. For that reason, they are more popular among Catholics and Orthodox than among evangelicals. The apostles were (probably) gone by the time these texts were written, but some kind of connection between them and the sub-apostolic writers cannot be ruled out. Clement of Rome may have been a contemporary of Paul and Peter, while Polycarp was supposedly a disciple of John. “Didache” claims that apostles still existed which, if true, would make the document very early. The Apostolic Fathers sound surprisingly familiar, since at least the Catholic and Orthodox Churches are based on traditions which in some sense go all the way back to this period. It's hardly surprising, therefore, that Catholic and Orthodox apologists use the Apostolic Fathers as evidence for Church tradition going all the way back to the original apostles and hence to Jesus himself.
Some familiar traits found in these texts include: Clement claiming that the bishop of Rome has authority over the local churches in Asia Minor, Ignatius describing a monarchic episcopacy and the absolute value of martyrdom, the “anti-Semitic” and heavily allegorizing tendencies in the Epistle of Barnabas, and the sacramental orientation found in the “Didache”. On other issues, the tradition is still not set in stone. Thus, “Didache” actually claims that the congregations have the right to test the apostles to see if they really are apostolic, which implies that ultimate authority is vested at the local level, perhaps with charismatic teachers? There is no clear exposition of the Trinity anywhere in the material, and the canon was still in flux. Thus, the author of 2 Clement actually quotes the Gospel of Thomas! It's also interesting to note that some of the Apostolic Fathers were themselves regarded as canonical by later generations of Christians.
How well does these texts reflect the original message of Jesus and the disciples? That is obviously a contentious issue. New religions can change surprisingly fast, as testified in the modern era by Mormonism, which changed on several fundamental points during Joseph Smith's own lifetime. Presumably, Christianity could therefore also have changed within a few decades after Jesus. In fact, it *must* have changed, otherwise Clement wouldn't be arguing with the elders of Asia Minor about who is in charge, and the “Didache” wouldn't be worried about who is and who isn't a true apostle. Note also that Ignatius fights “heretics” known as Docetists, apparently forerunners of the Gnostics – if the heresy is almost as old as the original message, where does that leave us? It's also intriguing to note that Ignatius claims to have a secret message which he refuses to impart to his flock, since they can't bear it yet. This suggests an oral tradition, perhaps a mystical one, running parallel with the written word. While both Catholics and Orthodox do claim that tradition could be purely oral, the idea of a *secret* tradition sounds less “orthodox”.
Already the New Testament records serious disagreements over the Christian message, including conflicts between Paul, Peter and Jacob. It's therefore a complicated task to ascertain what the “real” message might have been. Perhaps the question is wrongly put. Perhaps something really strange happened in Palestine circa AD 30, so strange that generations of mere mortals have been trying to come to grips with it for two millennia, without anyone completely succeeding...
The Apostolic Fathers is one link in that chain of attempted understanding.