This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
ShiKahr was a major city in the ShiKahr District, on the planet Vulcan. This city was built in a desert, on the edge of Vulcan's Forge. The L-langon Mountains were ten days travel, on foot, from this city. (TAS: "Yesteryear")
Future Starfleet officer and Federation Ambassador, Spock, was born in ShiKahr in January 2230. Sarek and Skon, Spock's father and grandfather, were also both residents of ShiKahr. (ENT: "Two Days and Two Nights"; TAS: "Yesteryear"; TOS: "Amok Time", "Journey to Babel"; Star Trek V: The Final Frontier; Star Trek Beyond)
In 2365, a holoprogram available to crewmembers of the USS Enterprise-D was "Shi-Kar Desert Survival, Vulcan". The program, with a code list number of 03-225, was listed in the exercise menu. (TNG: "The Emissary", okudagram)
Background information Edit
ShiKahr, as it was spelled in the script for "Yesteryear" as well as in the script for "Two Days and Two Nights" and in that actual episode, has also been spelled "Shi'Kahr", "ShirKahr", "Shirkar", and "Shi-Kar".
ShiKahr was shown in the background in the remastered version of "Amok Time".
The script and production report for the Star Trek: Enterprise episode "The Forge" (at StarTrek.com(X)) established that the United Earth Embassy was in Shi'Kahr. According to senior illustrator John Eaves, both the embassy and the Vulcan High Command building were supposed to be in the same city as each other.  Indeed, although the script of ENT: "Awakening" cited the location of the Vulcan High Command building merely as an unnamed "large Vulcan metropolis," the script of ENT: "Kir'Shara" identified that place – described on-screen as the capital city of Vulcan – as Shi'Kahr.
A deleted scene from Star Trek would have shown the birth of Spock in Shi'Kahr. Assuming that home was the same as their home in another scripted scene later, that scene would imply ShirKahr to be relatively close to Vulcana Regar, a distance that could be crossed quickly by hovercraft.
From the Star Trek Encyclopedia (4th ed., vol. 2, p. 272), "In the animated episode "Yesteryear", ShirKahr was seen to be Spock's home town. We saw ShirKahr in a new visual effects shot in the remastered version of "Amok Time" (TOS), in which its appearance was based on the animated episode. The location of the Vulcan High Command is from a scene description in the script for "The Forge". In "Kir'Shara", T'Pol described the city as the "capital". We intrepret it as meaning that it is the capital of the planetary government, since the Vulcan High Command was located there at the time. Shakier has also been spelled ShirKahr and Shi'Kahr. We assume that all are equally valid."