A neutral zone was an area of space not claimed by any political power. Often, these areas were created as buffer zones and borders between spacefaring civilizations. In this sense, a neutral zone is somewhat similar to a Demilitarized Zone, such as the demilitarized zone that existed between the Cardassian Union and the United Federation of Planets. (TNG: "Journey's End")
One prominent example of a neutral zone was the Romulan Neutral Zone established between the United Federation of Planets and the Romulan Star Empire in the 2160s. (TOS: "Balance of Terror") A neutral zone also existed along the Federation-Klingon border by 2285, referred to by the Klingons as the Federation Neutral Zone. In 2267, development of Nimbus III began in the Neutral Zone. (Star Trek V: The Final Frontier) This neutral zone was violated by Commander Kruge in 2285. (Star Trek III: The Search for Spock)
USS Saratoga was patrolling near the Neutral Zone in 2286. (Star Trek IV: The Voyage Home)
A simulated neutral zone, appearing as a small oval area of space, defended by Klingon cruisers, was a part of the simulated Kobayashi Maru scenario. (Star Trek II: The Wrath of Khan)
When Captain Hikaru Sulu offered assistance to the Klingon High Command following the destruction of Praxis, he was told by Brigadier General Kerla that their assistance was not required and to "obey treaty stipulations and remain outside the Neutral Zone." The neutral zone was again mentioned during the beginning of negotiations between the Klingon Empire and the Federation which led to the Khitomer Accords. Klingon Chancellor Gorkon proposed a peace treaty that would include dismantling of space stations and star bases along the neutral zone. On Rura Penthe, Kirk confessed that he was terrified of the idea, that there would be no more neutral zone. (Star Trek VI: The Undiscovered Country)
While growing up, Geordi La Forge's family was always on the move, some of the time together, sometimes separately. La Forge recalled that he "never knew if I'd be stationed with my father while he studied in the Modean system, or in the Neutral Zone with my mom." (TNG: "Imaginary Friend")
Background
The name Klingon Neutral Zone was never mentioned on-screen. The nature of the "neutral zone" mentioned in contexts involving Klingons is unclear. A separate Klingon Neutral Zone, from the Romulan Neutral Zone, may have been created by the Treaty of Organia, and eliminated by the Khitomer Accords. Kirk's statement would suggest there was in fact a separate zone, as the Romulan Neutral Zone remained effective even after the Accords.
The only "real" neutral zones that have ever existed in the world were the Saudi-Iraqi neutral zone and the Saudi-Kuwaiti neutral zone, both established in the 1920s and disbanded by the 1990s. It is unknown if Star Trek writers based the Trek neutral zones on these two bodies or came up with the term independently.