Vulcan finger-touching
Vulcan finger-touching was a ritualistic gesture among Vulcans involving the index and middle fingers. It was used throughout their culture, including in public as a greeting or to signify affection, intimacy, respect, or meditation. It was also used at reunions, at wedding ceremonies, and during the pon farr. Contact ranged from a simple two-fingertip touch to tracing around another's hand.

If a Vulcan was bonded with a member of another species, the non-Vulcan would adopt the practice. The Romulans also had knowledge of the ritual, and it could be used as a greeting or a sign of interest. It could also be used as part of a meditative practice to center and control one's emotions or to purge them. (; ; ; )

Notable occurrences
The instances in which Vulcan finger-touching was used included the following examples:

T'Pol and Koss during their wedding on  in 2154. ()

T'Pol and Koss upon greeting each other later that year. ()

Sarek of Vulcan and Amanda Grayson of Earth on a diplomatic mission aboard the  in 2268. ()

Spock and a female Romulan commander on her flagship later in 2268. ()

During Vulcan meditation, Spock used finger touching as part of his meditation practice to expunge his emotions through Kolinahr. ()

A young, revived Spock, who was entering pon farr, with Saavik on the  in 2285. Spock used the gestures she taught him while attempting to center his mind and emotions. ()

Hallucination 
While trapped in the Delta Quadrant in 2375, a telepathic pitcher plant made the crew of  believe they had returned to the Alpha Quadrant. Tuvok imagined that he was reunited with his wife T'Pel and greeted her in this manner. ()

Background information 
Regarding the origins of the Vulcan gesture involving touching of two fingers with two fingers, Leonard Nimoy explained in his book I Am Spock that the gesture was not meant to be the Vulcan equivalent of a Human kiss, but rather the Vulcan equivalent of holding hands in public: "The question came up as to what public sign of affection, if any, Sarek and his Human wife would display. Handholding was clearly out, but perhaps finger-to-finger contact of a ceremonial, dignified nature might work. Mark [Lenard] and Jane [Wyatt] took my comments to heart, and came up with the wonderful gesture where Amanda rests her first two fingers lightly upon Sarek’s two fingers. It worked beautifully, and added to the texture of [the episode]." Nimoy also described Vulcan finger-touching as "the beginning of the Vulcan mating ritual", "the Vulcan two-fingers-touching 'embrace'", and "the Vulcan version of foreplay". (I Am Spock, hardback ed., pp. 71-72, 237)

According to Harve Bennett, Leonard Nimoy imagined that, as a seven-year-old, Spock had engaged in some Vulcan finger-touching upon first meeting his betrothed, T'Pring. Nimoy related this concept to Bennett while they were writing . ()

While filming the Vulcan finger-touching with Saavik actress Robin Curtis for Star Trek III: The Search for Spock, Stephen Manley, portraying the seventeen-year-old Spock, grabbed Curtis' hands with both of his hands, an improvised gesture intended to portray passion. Leonard Nimoy, as the film's director, strongly approved of this moment, even fighting for it to remain in the movie when the executives at Paramount were eager to edit it out because it was eliciting a lot of laughs from viewers who weren't entirely familiar with Star Trek. Manley was extremely grateful for Nimoy insisting on the scene remaining. At photo opportunities during Star Trek conventions from  onward, Manley would typically hold his fingers touching those of female fans who wanted a picture taken with him. ("Spock: The Early Years", Star Trek: Motion Picture Trilogy (DVD)/(Blu-ray) special features)

A scene in the script of unmade movie Star Trek: The First Adventure involved Spock, as a cadet, teaching a stripper how to do Vulcan finger-touching. The script's stage directions identified this gesture as actually being pon farr.

In the first draft script of  (which had the working title "Equilibrium"), Vulcans T'Pol and Szon did some finger-touching during a dream T'Pol experienced. This finger-touching was to have taken place in 's mess hall, empty apart from the pair, and was described in the teleplay's stage directions thus: "Szon moves to her, reaching out two fingers in a traditional Vulcan gesture. T'Pol presents the back of her hand to him, and Szon presses his fingers against her flesh… T'Pol reacts to the sensual touch." In the dream sequence, the finger-touching preceded a very sensual mind meld between the couple.

The writers of the film  debated, among themselves, about the possibility of having  and  finger-touching in a turbolift scene from that movie. Ultimately, J.J. Abrams pointed out that such behavior would be unfamiliar and consequently puzzling to a new audience, so the pair of characters instead kiss in the same scene. In a Q&A, Roberto Orci and Alex Kurtzman agreed with this conclusion. 

Finger-touching
Communication