A psychological profile, or, more simply, psych 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 in the 23rd and 24th centuries.
In 2268, Captain Kirk asked Doctor Leonard McCoy whether Kirk's psychological profile indicated the possbility he was afraid of losing his job to the M-5 multitronic unit. (TOS: "The Ultimate Computer")
Later that year, when Kirk was showing signs of increased tension and emotional stress, he repeatedly resisted Dr. McCoy'sattempts 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")
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")