Ensure when you did a free background check, that that particular company was simply using your name and not your social security number.  Contact a private investigator to run a background check for you using his or her resources (many states now license private investigators).  If it comes up on those background checks, you may need to enlist the help of an attorney to help clear it.  If you feel you cannot afford the services of an attorney, at least consult with one (even if it is Legal Aid) so that if you attempt to contact the FBI or local police on your own, you are equipped with as much information as possible.  If the background check showed where this charge and conviction occurred, go to the court house and see if you can get some information or talk to the local prosecutor about the mistake and see if the prosecutor can help clear up the mistake.