Intuition, instinct or a gut feeling was information generated without conscious reasoning.
In 2259 of the alternate reality, James T. Kirk justified his decision to accompany Khan Noonien Singh and leave Spock behind, saying it was a gut feeling. (Star Trek Into Darkness)
According to James T. Kirk, no man achieved command in Starfleet without relying on intuition. Ultimately, it was this very thing that lead to the finding and destruction of the dikironium cloud creature (TOS: "Obsession")
In the last message Kirk recorded for Leonard McCoy and acting captainSpock, he asked Spock to rely on intuitive insight (claiming he knew he had it) or barring any ability to do so, ask McCoy for help. (TOS: "The Tholian Web")
When describing the limitations of the M-5 multitronic unit, Kirk said it didn't have intuition. (TOS: "The Ultimate Computer")
When Kirk met Areel Shaw in the bar on Starbase 11, he told her that he should have felt her presence in the air like static electricity. (TOS: "Court Martial")
While in the mines of Janus VI, Kirk asked Spock if his feeling that they were being watched was intuition. (TOS: "The Devil in the Dark")
Attempting to explain Trefayne's assertion that several hundred men with weapons were surrounding the citadel, Ayelborne claimed he was "quite intuitive" (TOS: "Errand of Mercy")
Jean-Luc Picard trusted Guinan's intuition enough to send the USS Enterprise-C back where it came from. (TNG: "Yesterday's Enterprise")
Geordi La Forge explained to Data that intuition was necessary because the facts were almost never entirely known. (TNG: "The Defector")
Data was incapable of intuition, but sometimes wished he had the capacity for it, since he felt it would be useful. (TNG: "Data's Day")
In 2374, after Captain Kathryn Janeway defeated Seven of Nine in multiple rounds of Velocity, Seven expressed that with her superior visual acuity and stamina, she should have won every round. Seven noted that Janeway had managed to acquire a target without even looking and Janeway told her that it was intuition. Seven described this as being as being a Human fallacy, but Janeway explained that she had unconsciously been aware of several factors, including the trajectory of the game disk after she had hit the wall, the sound it had made upon its return, and the shadow it cast on the hologrid. (VOY: "Hope and Fear")