Extrasensory perception (commonly abbreviated ESP) was a term used by 23rd and 24th century psychiatrists to describe the ability of humanoids to perform activities that require use of sensing capabilities beyond the five basic senses of sight, hearing, smell, taste, and touch.
In 2265 Starfleet believed that the ESP capacity in Humans was always quite limited, although it had been proven that some Humans could sometimes sense future happenings, read the backs of playing cards, see through solid objects, and cause fires to start spontaneously. There was controversy about the limits of such abilities as, despite such apparent evidence, Starfleet psychiatrist Dr. Elizabeth Dehner claimed that espers (those who exhibit ESP abilities) are "simply people with flashes of insight."
At that time Starfleet also had standardized tests to measure the aperception quotient (or "esper rating") of Humans to determine the degree of extrasensory perception present in an individual. Starfleet service records of the era included aperception quotient ratings.
Known Human contact with the galactic barrier (by the SS Valiant in 2065 and by the USS Enterprise in 2265) has resulted in significant augmentation in ESP abilities in those who already had a very high aperception quotient. Those whose esper rating was high but not extraordinarily so were injured by the contact, but not augmented. Later contacts with the Barrier in 2268 by the USS Enterprise did not result in any identified injury or augmentation. (TOS: "Where No Man Has Gone Before")
In the 24th century, according to his medical records and psychological profile, the Betazoid Tam Elbrun rated high on the ESP scale. The rating indicated he might be a prodigy, with his telepathic abilities swtiched on at birth as opposed to other Betazoids who developed this ability at adolescence. (TNG: "Tin Man")