Terrorist Apprehension and Record Retention Act of 2005 or the TARR Act of 2005 - Amends the Federal criminal code to provide that if the national criminal background check system indicates that a person attempting to purchase a firearm or applying for a State permit to possess, acquire, or carry a firearm is identified as a known or suspected member of a terrorist organization in records maintained by the Department of Justice or the Department of Homeland Security, including the Violent Gang and Terrorist Organization File: (1) all information related to the prospective transaction shall be automatically and immediately transmitted to the appropriate Federal and State counterterrorism officials, including the Federal Bureau of Investigation (FBI); (2) the FBI shall coordinate the response; and (3) all records generated in the course of the check that are obtained by Federal and State officials shall be retained for at least ten years.