You can submit an appeal form on the FBI website by visiting the National Instant Criminal Background Check area of the site and clicking on the Appeals link. From there, you need to choose whether you're appealing a firearm background check or a nuclear regulatory commission background check.
One you have selected the type of appeal you wish to initiate, a form comes onto your screen that you need to fill in and submit to the FBI as per the instructions on the form.
You can request an appeal if you believe the FBI has erroneously denied or delayed a firearm transfer based on a match to a record from the NICS. You can appeal the denial or delay of that decision. The FBI purges delayed decisions from the NICS within 88 days of the creation. The FBI recommends that you wait 30 days from the date you initiated the check prior to filing the appeal.
To appeal, you must provide your name, mailing address, and NICS transaction number or state transaction number. For delayed transactions, you must also submit a fingerprint card with the appeal request. The FBI recommends that you submit a fingerprint card with a denied appeal request.