What does 2 Corinthians 10:12 mean?
Paul is answering challenges to his authority as an apostle, made by false teachers among the Corinthians. One of their accusations was that, though Paul's letters were bold and strongly worded, he was unimpressive as a person when he was with them in person. In other words, the Corinthians did not need to worry about what Paul wrote since he could not back it up in a meaningful way when he showed up.Paul's answer is that he and his friends would, in fact, do everything he wrote in his letters, no matter how unimpressive he might appear.
Now he clarifies, somewhat sarcastically, that he does not see himself as being in competition with his opponents in Corinth. They are commending themselves to the Corinthians based on secular standards that included dramatic speaking ability and skillful self-promotion. Paul is not participating in the cultural competition to be the most popular or most followed public personality. In fact, he describes his opponents as being without understanding—they are unwise—as proven by their continual need to measure themselves against each other. That's a game Paul refuses to play. He doesn't need to win a cultural competition to prove he truly represents Christ to the Corinthians.