This guy is sort of a rival apostle to Paul in Corinth, though Paul doesn't really see things that way. He's still top dog in his—and God's—eyes.
In 1 Corinthians, Paul is pretty distressed at all the divisions that are cropping up in the church. Some of the Corinthians have even taken to calling themselves disciples of Apollos (1 Corinthians 1:12). Oh, that one's gotta hurt.
The odd part is that Paul doesn't seem to be mad at Apollos. Though his presence in Corinth is causing all kinds of tension, Paul never badmouths the guy. And it's pretty clear he doesn't lump Apollos in with the other false apostles. It's not really Apollos' fault that all this is happening. It's the stupid Corinthians. They just love to argue amongst themselves.
Paul stresses that he and Apollos are both "servants through whom you came to believe" (1 Corinthians 3:5). So, they're equal, right? Well, kind of. See, Apollos may be inspiring people in Corinth, but Paul is the head honcho. Paul "planted, Apollos watered" (1 Corinthians 3:6). Paul "laid a foundation [in the church], and someone else is building on it" (1 Corinthians 3:10). So basically, as much as people love Apollos, Paul is the founder and the one you should really be paying attention to. Hear that, Corinthians?
Apollos is working Paul's church, but he also doesn't seem to be under Paul's orders. Paul can send Timothy and Titus around from place to place running errands, but when he "urge[s Apollos] to visit with [the Corinthians…] he was not at all willing to come" (1 Corinthians 16:12). Looks like Apollos is a lone wolf.
The same Apollos shows up in Acts, too, where it's obvious he's also causing a little tension. Here Apollos "spoke with burning enthusiasm and taught accurately the things concerning Jesus" (Acts 18:25). But, when Paul's friends Prisca and Aquilla hear him, they take him aside and give him a couple pointers (18:26). It's clear that the Bible has a grudging respect for Apollos, but that everyone knows the dude is just plain in the wrong sometimes.