What is Paul relating to his first-century audience (1 Cor. 1:12-15)?
Answer: Paul is condemning any and all divisions among the early saints.
Some of these early faithful were ascribing to themselves what we might term: "Pauline", or "Apollonian", or "Cephite" Christians. He further admonishes his audience that it was Christ who was crucified, that in Christ we are baptized, that it is through Christ we are saved, and He alone.
While all of these men (Paul, Apollos, Cephas [Peter], and so on) were pillars in the first-century church, even they were not to be elevated above the status of Christ as simply Christians. It seems very likely these divisions could be extended to our day as well since there are many who adopt appellations that seem to fall into the same trap contrary to New Testament teaching.
We might also consider the same Letter of First Corinthians only two chapters later:
1 Corinthians 3:2b-4: "Indeed, even now you are not yet able, for you are still fleshly. For since there is jealousy and strife among you, are you not fleshly, and are you not walking like mere men? For when one says, “I am of Paul,” and another, “I am of Apollos,” are you not mere men?" (emphasis added).
Adopting the names of mere men (or anything else other than Christ) was especially problematic because we are informed of near-identical circumstances in two consecutive chapters.
Paul's message of unity must surely be one that extends to all generations, where we are simply to embrace the name of Christ, and never mortal men, irrespective of their "godly" status, just as those described in 1 Corinthians 1:12-15, and 1 Corinthians 3:2b-4.