Mojave was a town located in California on Earth. It was encircled by fifty miles of parkland, although it was considered not as nice as the parkland around bigger cities. In earlier times, the city was mostly a dry and arid region as it was situated within the southwestern portion of the Mojave desert. By the mid-23rd century, it was terraformed into a lush, fertile area.
During the early 23rd century, it was the birthplace of Starfleet officer Christopher Pike, who became captain of the starship USS Enterprise. When living in Mojave, Pike owned a pair of horses – Tango and Mary Lou – riding them into the groves with a picnic lunch every day. While in space, he missed his hometown, describing it as a "nice little town." (TOS: "The Cage")
In 2257, upon beaming over to the USS Discovery to take command of the starship, Pike introduced himself to Commander Saru by referencing his birthplace, relating that, in Mojave, he had learned "the best way to jump into a cold stream" was to "dive right in." (DIS: "Brother")
In the scripts for initial pilot "The Cage", the parkland near Mojave was described thus; "The locale is beautifully planted, rich park and forest land." Mojave itself was also detailed in "the far horizon where we can see a distant skyline of towering buildings with beautifully delicate architecture."
The Star Trek Encyclopedia (4th ed., vol. 2, p. 49) described Mojave as a southwestern region of the North American continent. The distant settlement was referred to as the "Mojave city" in this reference work.
The parkland in "The Cage" was filmed on Stage 16 of Desilu's Culver City lot.  A detailed background painting of a futuristic cityscape in a desert oasis was created for the parkland scene of "The Cage". It was later reused as a city on Planet Q in TOS: "The Conscience of the King". (The Art of Star Trek, p. 22; Star Trek Encyclopedia, 4th ed., vol. 2, p. 49)
Mojave was a somewhat prophetic choice for the home of space pioneer Pike, since, in 2004, some forty years after "The Cage" was written, Mojave became the USA's first inland spaceport, and the first to be cleared for horizontal spacecraft takeoff and launch.