The AFRCC (Air Force Rescue Coordination Center) is the office through which the majority, if not all of the search and/or rescue operations in and around the United States of America go through in some way or another. Consequently, their mission is "Continuously building a coordinated search and rescue network ensuring timely, effective lifesaving operations whenever and wherever needed."
To be more specific with the jurisdictional issues, the AFRCC deals with all of the SAR operations in CONUS (Continental United States), and occasionally, and if needed, provides assistance to Canada and Mexico. As you may or may not know, the US Air Force is separated into nine MAJCOMs (Major Commands). The AFRCC falls under ACC (Air Combat Command). I'm assuming they have their reasoning, but to me it seems a bit more appropriate to belong to, AMC (Air Mobility Command) or AETC (Air Education and Training Command), seeing as the Civil Air Patrol falls under AETC's jurisdiction.
Their motto, "These things we do, so that others may live." is in no way fictional, as the efforts of the AFRCC and its organizations have saved over 13,000 lives since its inception in 1974.
Some subordinate or cooperative SAR agencies to the AFRCC, whether they be civilian or military, include:
AFRCC Headquarters is located at Langley AFB, Virginia.
From the experience I have (I am a member of the Civil Air Patrol), SAR missions often involve an ELT (Emergency Locator Transmitter) or EPIRB (Emergency Position Indicating Radio Beacon) which is going off. Fortunately, the majority of signals are caused by a faulty unit, or a bumpy landing by the aircraft, or simply mishandling of the unit (I've found these things not in airplanes, but sometimes in warehouses, as in it was being shipped to be repaired, etc.). Nevertheless, lives are saved by the efforts of the AFRCC and its aligned agencies, since if a false signal was going off, it would be much harder to find the real one.
Sources: AFRCC Website (http://www2.acc.af.mil/afrcc//),
personal knowledge and experience