Clement of Rome was an elder in in the late 1st century. The Roman Catholics call him a pope (Pope Clement I), but his letter is one of the best evidences against the existence of a pope in the early 2nd century.
He wrote a letter to the Corinthians in A.D. 96, approximately 40 years after Paul wrote his two letters. He lets us know the Corinthians repented of their divisiveness, but now, a generation later, it was returning. They'd even defrocked a couple elders.
The letter is addressed from the Church of Rome. Clement's name is not on it. Since the 2nd century, however, Christians have been attributing it to Clement, and so it is known as First Clement.
It's a beautiful letter, full of the Spirit of early Christianity. While some of the New Testament writings were circulating at this time, you can tell from the letter that the Old Testament is still the main "Bible" that the Church had.
Irenaeus, almost a century after Clement of Rome wrote his letter, listed Clement as the 3rd bishop to succeed Peter in Rome. The previous two were Linus and Anacletus, who would thus be the 2nd and 3rd pope in Roman Catholic reckoning.
Much of Irenaeus' quote is given on the apostolic succession quotes page. There was clearly not a pope in the 2nd century church, but succession in Rome and other churches established by apostles was powerful evidence that the catholic churches were holding to the faith passed down by the apostles.
Now the faith which is in Christ confirms all these [admonitions]. For he himself by the Holy Spirit addresses us in this way: "Come, ye children, hearken unto Me; I will teach you the fear of the Lord. What man is he that desireth life, and loveth to see good days? Keep thy tongue from evil, and thy lips from speaking guile. Depart from evil, and do good; seek peace, and pursue it." (ch. 22, Scripture quote from Psalm 34)
Scholars think First Clement was attached to the Codex Sinaiticus, one of two complete Bibles dating from the fourth century. It's not in perfect condition, being 1700 years old, so there's pages missing.
There is another writing, called Second Clement, that is no longer attributed to Clement. Often now it will be found with the title, An Early Christian Homily. It is dated from a few decades after Clement died.
Clement of Rome is vilified here and there for believing that the story of the phoenix bird was a true story. He uses it as an example in nature of the resurrection. I can't imagine why it's a problem that he believed the poor science of his day.