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(covers information from several alternate timelines)
Praxis was a natural moon of the Klingon homeworld, Qo'noS. It was the only moon of that planet and was inhabited. Praxis was used as the Klingon Empire's key energy-production facility in the 23rd century. (Star Trek VI: The Undiscovered Country; VOY: "Flashback"; TNG: "Sins of the Father"; Star Trek Into Darkness, display graphic)
In 2293, Praxis was destroyed in a large explosion caused by over-mining and insufficient safety precautions. The explosion caused a powerful subspace shock wave that impacted on the USS Excelsior, a Federation starship outside the Neutral Zone heading back to Federation space under full impulse power after a three-year assignment in the Beta Quadrant.
Praxis' decimation resulted in a deadly pollution of Qo'noS' ozone, threatening the depletion of the oxygen in that planet's atmosphere within approximately fifty Earth years. Due to the Klingon Empire's enormous military budget, their economy did not have the resources with which to combat the catastrophe, forcing Chancellor Gorkon to approach the Federation with an overture of peace. This set into motion a series of events, eventually resulting in a lasting peace between the Federation and the Empire through the Khitomer Accords. (Star Trek VI: The Undiscovered Country; VOY: "Alliances", "Flashback")
The explosion of Praxis was loosely based on the Chernobyl disaster, one of several factors leading to the breakdown of the Soviet Union. (Captains' Logs: The Unauthorized Complete Trek Voyages, pp. 137-138) Likewise, the word "Praxis" is employed in the writings of Karl Marx, meaning active rather than merely theoretical socialism.
Praxis' destruction in Star Trek VI (particularly its placement at the start of the film) was influenced by a clap of thunder that begins Macbeth. In the playwright workshop at the University of Iowa, Nicholas Meyer (who co-wrote the screenplay of Star Trek VI and directed the film) learned it was a very effective method of attracting an audience's attention to the story. (audio commentary, Star Trek VI: The Undiscovered Country (Special Edition) DVD)
In a scripted line not included in the actual film, Praxis was to have been established as having no indigenous lifeforms.
In the script of "Flashback" (both the first draft and the final draft of the teleplay), Praxis was at first referred to as "the primary source of energy for the Klingon Homeworld." On screen, however, Tuvok instead says this line slightly differently, referring to Praxis as merely "a primary source of energy for the Klingon Homeworld," which is also how the closed captioning notates the line.
According to the reference book Star Trek: Star Charts (p. 61), Praxis was located in a separate system from Qo'noS. This system was closer to the Federation-Klingon border. In this system, Praxis was located in the Praxis system. The system's primary was a Class M star. The system was located in the Beta Quadrant.
In Star Trek Into Darkness, the Enterprise journeys to Qo'noS, and a large moon fragment strongly resembling the remains of Praxis, as depicted in Star Trek VI, can be seen. Co-writer Alex Kurtzman stated, "It certainly was inspired by Praxis, however, it's not specified, so we're leaving it open as to exactly what that moon was."  Roberto Orci agreed in a comment on Twitter, stating that the moon was "not specified as Paraxis [sic]. But could be."  This destroyed sole moon of Qo'noS is, however, identified as Praxis on screen, in a table of information. 
According to the novelization of Star Trek Into Darkness, the broken moon orbiting Qo'noS was Praxis. According to the comic book prequel Star Trek: Khan, this broken moon is the result of a mission to destroy Praxis by John Harrison for Section 31 (who learns his true identity as Khan while on said mission).
In the Star Trek: Myriad Universes story The Chimes at Midnight, set in an alternate timeline (specifically, one from TAS: "Yesteryear" in which Spock died as a child), Praxis is chosen by Thelin in 2290 as the test site for the Genesis Device (which in this timeline had led to full-scale war), both as a demonstration of force and because, with nearly all of its inhabitants living in areas either underground or shielded, there would be little or no loss of life. The action forces the Klingons to surrender; unfortunately, the Genesis Wave overwhelms most shields and reaches below the ground, and of the population of half a million, only 2,000 survive. However, during peace negotiations, it is discovered that Genesis actually saved the moon, as the explosion which occurred in the prime universe had been inevitable within a year before the wave had strengthened the planet and put an end to the mining that would eventually destroy it.
In the novel Sarek by A.C. Crispin, in addition to polluting Qo'noS' atmosphere, Praxis' explosion has created a ring of debris around the planet that subjects the surface to constant, and sometimes deadly, meteor showers, which also makes space traffic to and from the planet extremely dangerous – another reason why the planet is becoming uninhabitable.
In Star Trek Online, Praxis is presented as the former moon of Qo'noS. After events that occur prior to the year 2409 broke the Khitomer Accords, mining on the remnants of Praxis renewed. The remnants of Praxis orbiting Qo'noS are collectively referred to as the Praxis Belt.