This poor guy doesn't even get a name, but he does get a heaping helping of scorn from Paul.
It seems that on Paul's second visit to Corinth—which he calls the "painful visit"—a fight breaks out. Some guy in the church has done something. It's bad. Paul is offended by it. But the Corinthians, bless their little hearts, aren't bothered at all.
Because of this "offending brother," the visit does not go well at all (2 Corinthians 2:1-2). Paul then writes a long "tearful letter" to the Corinthians asking them to just punish this guy and mend their fences with him (2 Corinthians 2:4). The Corinthians agree and all is well. Ice cream sundaes for everyone!
So what happened to the Offending Brother? He was punished by a majority in the church for his crime (2 Corinthians 2:6). What was the punishment? We don't know. What was the crime? No clue. Did the punishment fit the crime? Well, now you're just being silly. The point is that Paul doesn't see fit to mention any of these details (maybe he doesn't want to open old wounds). So we have no way of knowing what exactly went down. Paul is just focusing on the positive here and not dwelling on what the Offending Brother did wrong.
And, to Paul's credit, whatever the Offending Brother did, Paul is ready to forgive him. Apparently, the punishment is good enough and the Offending Brother can now be welcomed back into the fold (2 Corinthians 2:7-8). Forgiveness is wonderful, isn't it?