No one knows for sure what Paul's thorn in the flesh was. There have been various theories offered, such as a chronic sickness and bad eyesight. We just don't know because the Bible does not tell us. It doesn't mean we can't offer some educated guesses, but first let's take a look at the relevant Scripture.
Paul said in 2 Corinthians 12:7-8, "Because of the surpassing greatness of the revelations, for this reason, to keep me from exalting myself, there was given me a thorn in the flesh, a messenger of Satan to torment me-to keep me from exalting myself! 8 Concerning this I implored the Lord three times that it might leave me." This problem may very well have been the same one he mentioned in Galatians 4:13-14, "but you know that it was because of a bodily illness that I preached the gospel to you the first time; 14 and that which was a trial to you in my bodily condition you did not despise or loathe, but you received me as an angel of God, as Christ Jesus Himself." Neither verse tells us what the issue was, but the affliction may have been painful as could be designated by the word "thorn." But then again, if it were an issue with his eyesight, that could be considered a thorn in the flesh that was not painful. Let's take a look at that option.
In Acts 9:3 Paul says, "As he [Paul] was traveling, it happened that he was approaching Damascus, and suddenly a light from heaven flashed around him." Because of this light, Paul fell to the ground and was blinded for three days (Acts 9:8-9; 22:11). Also, Paul says something very interesting in Galatians 4:15, "Where then is that sense of blessing you had? For I bear you witness that, if possible, you would have plucked out your eyes and given them to me." Could it be that because of the encounter with Christ Paul had been blinded, and this is why the Galatians wanted to, so to speak, pluck their eyes out for him? It seems like a real possibility.
At any rate, Paul had asked three times to be delivered of affliction but God did not heal him. The reason is obvious, as is stated by Paul himself, that it was to keep him humble (2 Cor. 12:7). It is worth noting that our afflictions which are permitted by God are often for the same reason. We cannot see how pervasive sin really is in us, and sometimes we're humbled by our afflictions and return to God and trust him. This, of course, is a good thing.