In papal lists he is sometimes placed second after Peter, sometimes third after Linus, and sometimes fourth after Linus and Anacletus; according to Tertullian, at least, Peter consecrated all three men as bishops to care for the community or communities at Rome so that he could devote himself to preaching, and Clement was the one with the most responsibility while Peter was alive. In any case, the papal lists usually follow St. Irenaeus, and that list puts the order as Peter, Linus, Anacletus, Clement. From his letter to the Corinthians (Chapter 38):
Let our whole body, then, be preserved in Christ Jesus; and let every one be subject to his neighbour, according to the special gift bestowed upon him. Let the strong not despise the weak, and let the weak show respect to the strong. Let the rich man provide for the wants of the poor; and let the poor man bless God, because He has given him one by whom his need may be supplied. Let the wise man display his wisdom, not by [mere] words, but through good deeds. Let the humble not bear testimony to himself, but leave witness to be borne to him by another. Let him that is pure in the flesh not grow proud of it, and boast, knowing that it was another who bestowed on him the gift of continence. Let us consider, then, brethren, of what matter we were made, -- who and what manner of beings we came into the world, as it were out of a sepulchre, and from utter darkness. He who made us and fashioned us, having prepared His bountiful gifts for us before we were born, introduced us into His world. Since, therefore, we receive all these things from Him, we ought for everything to give Him thanks; to whom be glory for ever and ever. Amen.
First Clement, as it is usually called, was one of the candidates for books to be part of the New Testament; this seems to have arisen because the Corinthians began to read it in their churches after having received it from Clement, and the practice spread to other churches influenced by the Corinthians. Evidence of its authenticity is quite good. The best estimates for the date it was written place it in the last decade of the first century, making it roughly contemporary with the book of Revelation.
According to a (very late) legend he was martyred by being tied to an anchor and thrown into the sea, so the symbol associated with him is the anchored cross, also known as a St. Clement's Cross. He is sometimes identified with the Clement of Philippians 4:3, and very often with the Clement mentioned in the Shepherd of Hermas. He is also sometimes said to have been a freed slave. While the testimony that he was a contemporary of the apostles is universal, what we know about his life is practically nil, beyond what we can glean from his one surviving authentic text and his appearance on the succession lists for the episcopate of Rome.