(covers information from several alternate timelines)
Tribbles (Polygeminus grex) were small, non-intelligent lifeforms originating from Iota Geminorum IV. Known for their prodigious reproductive rate, these round, furry creatures emitted cooing sounds while touched, which had a tranquilizing effect on the Human nervous system.
On their own, tribbles are quite incapable of directly harming sentient beings - they do not even possess "teeth" as such. They are also fairly slow. Nonetheless, if their population growth is left unchecked they can rapidly multiply and devastate entire ecosystems on a planetary scale. (TOS: "The Trouble with Tribbles"; DS9: "Trials and Tribble-ations")
Fully half of a tribble's entire metabolism is devoted to reproduction. Born pregnant, a single tribble with sufficient food can quickly increase its number exponentially through asexual reproduction, bearing an average litter of ten every twelve hours. Tribbles are hermaphroditic, possessing both male and female reproductive organs, which makes them capable of self-fertilizing at will. Given sufficient food supply, a single tribble can give rise to progeny numbering over one million in less than four days.
Tribbles were first encountered by Humans in the early 2150s when Denobulan doctor Phlox brought a small number aboard Enterprise NX-01 as an easily sustainable food source for his pets. (ENT: "The Breach")
The tribble was later encountered by the Federation aboard Deep Space K-7 in 2268, when Cyrano Jones brought a small number aboard to trade. However, a single tribble brought aboard the USS Enterprise quickly multiplied. The tribbles were instrumental in foiling a Klingon plan to poison a shipment of quadrotriticale intended for Sherman's Planet, when those that had fed on the grain were found dead. (TOS: "The Trouble with Tribbles")
At the same time, the crew of USS Defiant, transported through time by Arne Darvin, prevented him from using a tricobalt device hidden inside a tribble to kill James T. Kirk. (DS9: "Trials and Tribble-ations")
Tribbles were an ecological menace for the Klingons, to whom a tribble would react violently. The Klingons first attempted to breed a predator to eliminate the threat. The glommer was used only once, in 2269, aboard the USS Enterprise, and proved unable to deal with the large number of tribbles aboard. (TAS: "More Tribbles, More Troubles")
During this time, Cyrano Jones attempted to genetically engineer the tribbles so that they would not reproduce. As a result of his dabbling, he deemed them to be "safe" so as to be compatible with humanoid ecologies, making them "great pets and profitable, too." Unknown to Jones, at the time, and as was later discovered by Doctor Leonard McCoy, Jones' genetic engineering was described as being "very slipshod," as the resulting tribbles appeared to grow in immense size, instead of rapid breeding. McCoy later discovered that the so-called "giant tribbles" really were not as they appeared, but rather colonies of tribbles. McCoy was able to remedy the situation by giving them "a simple shot" of neoethylene, which caused the tribble colonies to break down into their individual units with a slower metabolic rate, and finally becoming "safe tribbles." (TAS: "More Tribbles, More Troubles")
The Klingons' solution to the tribble menace was to wipe the species out entirely. Klingon warriors were sent to hunt them down throughout the galaxy, and the tribble homeworld was obliterated in the late 23rd century. When told of this, Odo dryly remarked, "Another glorious chapter in Klingon history. Tell me, do they still sing songs of the Great Tribble Hunt?" (DS9: "Trials and Tribble-ations")
In spite of the destruction of the tribble homeworld, many Humans continued to have the small creatures as pets from the late 23rd century, well into the late 24th century, as one child aboard the USS Enterprise-D appeared to have one as a pet in 2371. (Star Trek III: The Search for Spock; Star Trek Generations)
Information about tribbles was displayed by the computer in Keiko O'Brien's schoolroom on Deep Space 9 in 2369. (DS9: "A Man Alone", "The Nagus") Tribbles were displayed in an LCARS graphic in one of the USS Enterprise-D's science laboratories when the ship crash-landed on Veridian III in 2371. (Star Trek Generations)
Tribbles were reintroduced into the 24th century when Captain Benjamin Sisko and his crew brought at least one of the creatures back to 2373, following a time-traveling visit to the tribble-infested space station K-7 of 2268. The Promenade and Quark's Bar on Deep Space 9 were subsequently overrun with the creatures. According to Captain Sisko, the Federation's Department of Temporal Investigations was not told about the reintroduction of the tribbles because "they didn't ask." (DS9: "Trials and Tribble-ations")
Alternate reality Edit
A year later, Dr. Leonard McCoy injected the blood of Khan into the necrotic tissue of a dead tribble, in order to test the regenerative properties of the blood. A little while later, shortly after the death of James T. Kirk, Khan's blood resurrected the tribble in front of a stunned McCoy. McCoy then ordered Khan captured alive so that his blood could be used to the same effect on Kirk. (Star Trek Into Darkness)
Background information Edit
In the audio commentary for "More Tribbles, More Troubles", David Gerrold explains that the idea for tribbles came from a fuzz-ball keychain owned by a college girlfriend of his, named Holly Sherman (for whom Sherman's Planet is named). He also mentions that Wah Chang hired a woman named Jacqueline Cumere to make five hundred tribbles for "The Trouble with Tribbles", and explains the procedure for making them.
Tribbles are made from a piece of "figure-eight shaped" fake fur, sewn up to make a ball about five inches in diameter. The ball is filled with foam rubber; some tribbles were made by sewing a decapitated walking toy dog inside the fur skin, and the toy activated to make the tribble crawl, as revealed in a DVD text commentary by Michael and Denise Okuda for "The Trouble with Tribbles".
Sound Editor Douglas Grindstaff was involved in creating the noises made by the tribbles. Producer Gene Roddenberry first showed Grindstaff a tribble prop and they discussed how they should tackle the creation of the different sounds made by the creatures. There had to be sounds for their various moods, such as for a very comfortable tribble as well as a fearful or disapproving tribble, rearing up and away from someone it didn't like. There also had be sounds for different quantities of tribbles. "I found a dove 'coo,' I flipped the track over, and shaved off part of it with a razor blade," Grindstaff recollected. "I then made a loop out of it, put it on a variable speed machine, and changed the pitch of it to different frequencies. Then I chose those frequencies I liked the best and decided which ones I would use in each spot. I then went to a screech owl for the sound of the tribble rearing up, took the screech owl and played with it and got variations of it. Then I took a bunch of little balloon sounds, where you'd get a balloon and rub it, and we mixed that with other little squeaky sounds that we had in the library, playing with all these at variable speeds, mixing them and making new sounds out of them, and putting them all together until we came up with the various composite sounds used throughout the whole show." (The Star Trek Interview Book, p. 236)
Following the making of "The Trouble with Tribbles", tribble props were repeatedly found around the set of Star Trek's original series. Commented DeForest Kelley, "We'd had these tribbles everywhere. They had them in the prop room and they kept showing up here and there for weeks after we filmed the episode. A tribble might show up as an extra breast on Nichelle [Nichols] or something like that. I guess everybody was still kind of tribble happy and every time we got to work a tribble in somewhere, we were working them in. Instead of pulling out a communicator, somebody would pull out a tribble. They'd continually pop up somewhere. Pull out a drawer and somebody would reach for something and there'd be a tribble there [....] It took weeks to get all the tribbles off the set." Kelley also remembered that – in a scene of an episode that he reckoned was "probably" "Journey to Babel", where his character of Dr. McCoy was meant to be making a surgical incision in another character – he (as Dr. McCoy) apparently removed a tribble from the patient after making the incision, an action that provoked much laughter on the set. (The World of Star Trek, 3rd ed., p. 86)
Having appeared as Klingon captain Koloth in "The Trouble with Tribbles", actor William Campbell was made aware of how extremely popular the tribbles were with fans. "The actors weren't really the stars of the show, those little animals were," Campbell opined. "They really intrigued whoever watched it. One little girl even made a tribble and sent it to me." (The World of Star Trek, 3rd ed., p. 121)
Another homemade tribble was created for Terry Farrell by her grandmother, during her childhood, long before she portrayed Jadzia Dax in Star Trek: Deep Space Nine; a frequent viewer of TOS, Farrell kept her tribble toy in her bedroom, by a television set. (The Official Star Trek: Deep Space Nine Magazine, Vol. 2, p. 56)
Director Joseph Pevney once speculated that, if tribbles had been proposed to make their debut appearance during the third season of the original series rather than the second, the idea would have been rejected, due to the regular cast becoming increasingly concerned about being seen as the heroes of the series. Pevney argued, "The hero of the show was a little fuzzy animal, and they don't want that. They want to constantly be the heroes." (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 48)
In "More Tribbles, More Troubles", the tribbles appear pink in color. Unknown to the rest of the production staff, the director, Hal Sutherland, was colorblind, so to him, pink was light gray.
Harve Bennett once characterized the tribbles in Star Trek III as "Christmas stocking presents to the fans" and went on to say, "That's, 'Hey, kids, you came to see the movie, here's a treat, it's our joke.'" (The Star Trek Interview Book, p. 271)
An edible tribble was designed to be ripped open as a snack by one of the Klingon guards on Rura Penthe in Star Trek VI: The Undiscovered Country, filled with gummy worms and tapioca "guts," but it was not seen in the movie, as it was thought to be too disgusting. A scepter that Rura Penthe's commandant carries has fur that was scripted to be from a tribble. (The Making of the Trek Films, 3rd ed., p. 129)
Some of the most memorable ways in which tribbles are featured in "Trials and Tribble-ations" were devised by DS9 writing staffer Ronald D. Moore. These ideas included tribbles being tossed out of the storage compartment bin from "The Trouble with Tribbles" by one of the DS9 crew – hitting Kirk on the head with them – and another tribble actually being what is more-or-less a disguised bomb. (Star Trek: Deep Space Nine Companion, p. 384)
In the first draft script of "Trials and Tribble-ations", a hypothetical evolutionary link was made between tribbles and Trill symbionts, as some xenobiologists had apparently theorized that tribbles were "distant evolutionary cousins to Trill symbionts."
The tribble props for "Trials and Tribble-ations" were obtained from Lincoln Enterprises. Prop master Joe Longo later recalled, "We bought 1,400. .... [They] only had about 600 there and had to make up some more – it took two weeks. And then we had to make up some in Drapery; we didn't have enough of the large ones." (Star Trek: Communicator issue 110, p. 55)
DS9 special effects supervisor Gary Monak and his team of live effects artists at Paramount were tasked with matching the actions of operable tribble props in "Trials and Tribble-ations" with the equivalent props of thirty years beforehand. This was despite the fact that the DS9 episode features more moving tribbles than there are in "The Trouble with Tribbles". Reported Monak, "We're having about 50 different moving ones made up; they're all pretty much like crawlers or wigglers, the one that's supposed to jump when it sees Klingons is a remote control one we've made." The workings inside the latter prop consisted of a variable-speed wheel with an off-center gearing. (Star Trek: Communicator issue 110, p. 58)
Worf actor Michael Dorn hated tribbles, much like how his own Klingon character felt about the species. "It's just this whole cute thing. It just drives me up a wall," Dorn expressed. (Cinefantastique, Vol. 32, Nos. 4/5, p. 93)
Tribbles were mentioned in a 2004 treatment in which J. Michael Straczynski and Bryce Zabel outlined their plans for a reboot Star Trek TV series that ultimately went undeveloped. In the document, the tribbles were once again depicted as encountering the Enterprise, Captain Kirk, and the rest of the ship's crew, though in a new story. The treatment also pondered about the creatures, "What if they came equipped with an agenda, an attitude...and teeth?" 
The tribble in the film Star Trek was operated by remote control, used by a production staffer standing just out of the camera's line of sight. (Star Trek Special Edition/Three disc Blu-ray documentary featurette "To Boldly Go") Similarly, the tribble in Star Trek Into Darkness was a puppet operated by Director J.J. Abrams, from under a table which the tribble sat on. (enhanced commentary, Star Trek Into Darkness & Star Trek: The Compendium DVD special features)
Two tribble fur balls were sold together in the 40 Years of Star Trek: The Collection auction for US$4,800. The lot had been estimated to bring in only between US$800 and US$1,200. These tribbles were made for the DS9 outing "Trials and Tribble-ations" and were also used in the bazzar scene in ENT: "Rajiin".
According to the Star Fleet Medical Reference Manual, the mutual dislike between Klingons and tribbles has to do with the fact that both species have keen senses of smell, and apparently, each found the "stench" of the other extremely unpleasant. Tribbles also found food using their sense of smell.
In the novelization of The Wrath Of Khan, when McCoy gives Kirk a birthday present, Kirk (under the influence of Romulan ale) asks, "...is it a tribble?!"
In the TNG novel Q-in-Law, Q taunts Worf by offering him a tribble as a present. Worf shares most Klingons' instinctive loathing of the creatures, a fact later confirmed in "Trials and Tribble-ations".
In the DS9 novel The Big Game, Odo does not allow a tribble to be brought onto Deep Space 9 until it is confirmed to be sterile.
In Star Trek Online, the tribbles' tranquilizing effect on the nervous system is used as a healing factor during gameplay. They can also be bred by introducing them to various foodstuffs, giving the player access to different subspecies of tribbles that bestow different benefits. They also seem to spontaneously breed if there is at least one in the player's inventory and a food source is available. Special tribbles have become a common reward for either testing major updates (called seasons) or, more recently, logging onto the game near the beginning of a season. These tribbles typically give buffs against a major enemy of the season. The game's test server is named Tribble.