Yes, NCIC does belong to the FBI.
http://www.fbi.gov/hq/cjisd/ncic.htm
Any law enforcement agency in the US can do an NCIC (FBI) background check, but these are mainly for law enforcement purposes, with a few exceptions. (Public housing is allowed to do them to a limited extent. There are probably some others I'm not aware of.)
We used to do firearms purchases during the dark days of the Brady Bill when firearm purchases went through the sheriff's department (where I worked) and there was that two day waiting period. This changed later (thank God!) to the central number which FFL holders now call for background checks.
I don't know what that central number is or who it belongs to, but I assume it's the NCIC division of the FBI.