The psych test was a trial undergone by prospective Starfleet cadets during the Starfleet Academy entrance exam. TAC officer Chang described it as "a psychological evaluation based on reactions to various individual problems." It was designed to force an applicant to face their greatest fear and evaluate their reaction to it.
In 2364, Wesley Crusher underwent the psych test as part of his entrance exam on Relva VII. He was put in a situation in which it appeared that two crewmen were about to die in an imminent explosion, and Wesley could only save one of them. It was determined that Wesley feared this situation because, years earlier, Captain Picard had to make a similar choice which resulted in the death of Wesley's father. (TNG: "Coming of Age")