A background check was an inquiry into an individual or starship's history for investigative purposes.
In 2266, James T. Kirk attempted to confirm Doctor Thomas Leighton's suspicions that Governor Kodos and Anton Karidian were one and the same by asking the USS Enterprise's computer for the latter's background. (TOS: "The Conscience of the King")
In 2367, Admiral Norah Satie ordered a background check into Simon Tarses' friends at the Starfleet Academy training program, following the revelation that he was part Romulan, not Vulcan, as he'd claimed. (TNG: "The Drumhead")
Later that year, Major Kira Nerys was interrupted in the middle of running a background check on the Valerian starship Sherval Das by Sisko, who allowed the ship to dock at Deep Space 9. Kira was confident that she could prove that they were running weapons-grade dolamide. (DS9: "Dramatis Personae")
In 2372, Odo checked the background of both the captain and the passengers of a Klingon civilian transport ship which had been destroyed by the USS Defiant. Those background checks led to the discovery of a Klingon conspiracy to discredit Worf. (DS9: "Rules of Engagement")