Perhaps all we know about Cotton-eyed Joe is what we don't know about him: we neither know where he came from, nor where he went. We do know, though, that had it not been for him, we would have been married a long time ago.

In all seriousness, he is just a character from a song who stole the heart of the narrator's girlfriend. The 1994 Rednex version of the song offers little more explanation of who he was than this, but earlier versions of the tune, dating back to slave songs in the antebellum south, describe Cotton-eyed Joe as cross-eyed, flat-nosed, and toothless, but also tall and slim. He's said to have taken the girl to Tennessee, answering the Rednex's second question about Joe.

As for what "cotton-eyed" means, nobody is entirely sure. It may be that Joe went blind from drinking bad moonshine, that his eyes were milky-white from some sort of an infection, that he had cataracts or glaucoma, or merely that he had very light colored eyes, among many other explanations.

Sources:

http://www.traditionalmusic.co.uk/negro/folk-songs%20-%200169.htm

http://www.ibiblio.org/fiddlers/CORK_COT.htm#COTTEN-EYED_JOE_%5b1%5d