Actually, Jack's last name is in fact Teague. Jackson Edward Teague is his real name. Though, as you may already know, Jack isn't too fond of Teague, and you can't have TWO Captain Teagues. So, to avoid association with his dad, he personally goes by Sparrow. It makes sense if you think about it.
actualy, im pretty sure that teague is only "theoreticaly" his father. i mean when has he actualy admited they were related? did i miss a scean or something?!? other than that fact that they dress alike and talk alike there realy is no evidence that says they are related....other than that "how's mum" part.