A psychological profile was a scientific analysis of a person's mental health and ability. Such profiles were routinely kept on applicants to and members of Starfleet during the 24th century.
In 2268, when Captain Kirk was showing signs of increased tension and emotional stress, he repeatedly resisted Doctor McCoy's attempts to run a psychological profile on him. Unbeknownst to McCoy, Kirk was working under Federation orders to steal a Romulan cloaking device. (TOS: "The Enterprise Incident")
In 2364, the "psych test", a portion of the Starfleet Academy entrance exam, used information from applicants' psychological profiles to construct a situation involving the applicant's greatest fears. (TNG: "Coming of Age")
Commander Riker reviewed Lieutenant Reginald Barclay's psychological profile in 2366 which indicated that he had "seclusive tendencies." (TNG: "Hollow Pursuits")
While investigating the suicide of Lt. Daniel Kwan in 2370, Counselor Deanna Troi found that Kwan had an exceptionally stable psychological profile. (TNG: "Eye of the Beholder")
While visiting Doctor Julian Bashir on Deep Space 9, Doctor Lewis Zimmerman started building a complete psychological profile of Bashir for use in his Long-term Medical Holographic program. Building a complete profile involved interviewing Bashir's closest friends and family. (DS9: "Doctor Bashir, I Presume")