(covers information from several alternate timelines)
Kolinahr (ko-li-naar), as a word, described both the Vulcan ritual by which all remaining vestigial emotions were demonstrated as purged, and the mental discipline whereby this state was subsequently maintained. Not all Vulcans were required to make this final journey to pure logic. The duration of Kolinahr training was quasi-monastic in nature, and could vary from two to six (or more) years. (ENT: "The Andorian Incident"; Star Trek: The Motion Picture; VOY: "Flashback", "Fury")
The Vulcan monastery at P'Jem was a remote sanctuary for kolinahr and peaceful meditation. According to Sub-commander T'Pol, visitors would be turned away during the time of kolinahr. (ENT: "The Andorian Incident")
In the alternate reality, Spock discussed the Kolinahr ritual with his mother shortly before making the decision to enter Starfleet instead of the Vulcan Science Academy. He wanted to ensure that if he decided to undergo Kolinahr, she would not interpret his decision as a rejection of her, or his Human half. She lovingly reassured him that, regardless of the decision he made, she would proudly stand behind him. (Star Trek)
In the prime reality, Spock began the process of the Kolinahr ritual after retiring from Starfleet circa 2270. By the end of his 2270s stay on Vulcan, Spock had completed all but the final ritual of the Kolinahr, but failed to do so after receiving powerful telepathic signals from the V'ger entity in deep space, and experiencing strong emotions as a result.
In 2298, Tuvok began immersing himself in Kolinahr after retiring from Starfleet. Six years later, he entered into Pon farr and married T'Pel. After they choose to raise a family, Tuvok postponed his studies. (VOY: "Flashback")
Spock informs Dr. McCoy of his intention to enter Kolinahr at the conclusion of The Lost Years, just after the Enterprise concludes its original five-year mission. Spock predicts that he will never see McCoy, Kirk, or any member of the Enterprise crew again, a prediction that McCoy instinctively refuses to accept.
Forgotten History briefly looks at Spock's reasons for going through Kolinahr, as he feels that his remaining emotions are the reason that he has been mentally compromised by so many creatures during the Enterprise's voyages.
The events pictured in Troublesome Minds and Crisis of Consciousness are also given as motivating factors for Spock's decision; at the end of the latter novel, he requests all data from the Vulcan Science Academy on the discipline.
In The Fire and the Rose, Spock again seeks to achieve Kolinahr when faced with a loss of emotional control after Kirk's apparent death on the USS Enterprise-B. He succeeds, but following his mother's death and a confrontation with Leonard McCoy, Spock undergoes a risky procedure to reverse the Kolinahr and restore his ability to feel emotions as he realizes his inability to grieve for his mother.
In the novelization of the film, in the scene where Spock fails to complete his Kolinahr, it is stated to be Kirk's mind he is telepathically connecting with, not V'ger's.