An obsession, fixation, or fetish was a preoccupation with an object, person or task to the point of irrational behavior.
Obsessions could take many forms, such as conspiracy theories, survivor guilt, and addiction.
In the 1940s, the sport of baseball was a "national obsession" in the United States of America. (TNG: "The Big Goodbye")
In 2152, the crew of the Enterprise began exhibiting obsessive behavior due to radiation emitted by a trinary system, becoming inordinately concerned with trivial matters. T'Pol was the only member of the crew unaffected due to her Vulcan physiology and with aid from Captain Jonathan Archer navigated the vessel away, which resolved the problem. (ENT: "Singularity")
In 2268, the crew of the USS Enterprise encountered a Dikironium cloud creature that, eleven years earlier, claimed the lives of 200 crewmembers of the USS Farragut. Captain James T. Kirk, a lieutenant then, suffered from survivor guilt, believing that had he attacked the creature when ordered, the crew would have survived. Kirk's actions hunting down the creature cost the lives of a number of crewmembers, but upon learning phaser fire was ineffective against it and thus attacking when ordered wouldn't have changed the outcome, Kirk came down from his drive and took a more logical means to end its threat. (TOS: "Obsession")
Sometime in the 2270s, Leonard McCoy accused Kirk of being obsessed with USS Enterprise, so much so that he used the V'ger situation to regain command of the ship and was therefore blinded to more important responsibilities. He used Kirk's reaction to Will Decker as evidence, which McCoy felt to be inappropriate to the situation. (Star Trek: The Motion Picture)
In 2364, Hagon asked why Lutan was so obsessed with Lieutenant Tasha Yar over Lutan's First One Yareena who could offer him much more and owned many things and many lands. (TNG: "Code of Honor")
That same year, Captain Jean-Luc Picard claimed that a lot had changed in the past three-hundred years, that people were no longer obsessed with the accumulation of things. (TNG: "The Neutral Zone")
In 2370, Deanna Troi described the Iyaaran ambassador Loquel as being obsessed with food, chocolate in particular. William T. Riker replied that she must therefore be "in heaven", but Troi admitted that his interest extended beyond hers. (TNG: "Liaisons")
In 2375, Doctor Julian Bashir speculated that the choice of a TR-116 rifle in the killings of several residents of Deep Space 9 might have been due to a psychological obsession or fetish on the part of the murderer. (DS9: "Field of Fire")
Obsessions could cause acute anxiety, sleeplessness, and paranoia. In 2376, Reginald Barclay asked Counselor Deanna Troi for help after he believed he was becoming obsessed with USS Voyager (to the point of spending an inordinate amount of time with holographic versions of the crew). After being removed from duty, he changed his mind, arguing that the obsession helped him perform his duties better and asking Troi whether his mental state was more important than Voyager. When she reminded him that he was important too, he finally admitted that it was a coping mechanism following his transfer from USS Enterprise-D, whose crew he had regarded as family. (VOY: "Pathfinder")