(covers information from several alternate timelines)
This page contains information regarding Star Trek: Lower Decks, and thus may contain spoilers.
Prior to 2269, the Kzinti fought four wars with Humankind, and lost all of them. In the course of these conflicts, some Kzinti fed on Human meat. The last such war took place in the late 21st century. By the Treaty of Sirius, the Kzinti were forbidden all weapons apart from police vessels. (TAS: "The Slaver Weapon")
In 2269, Dr. Keniclius 5 cited the need to subdue the depredations of hostile species including the Kzinti as his justification for planning to create a "master race" consisting of an army of giant Spock clones. (TAS: "The Infinite Vulcan")
Later in 2269, Kzinti archaeologists discovered two Slaver stasis boxes on Kzin. One of these boxes was turned over to Starfleet. The Kzinti kept the other for themselves, only to find it empty. With the secret support of the Kzinti government, a plan was devised to reacquire the former box from Starfleet's custody, one which would be disavowed by the Highest of Kzin in the event of its failure.
On stardate 4187.3, as the USS Enterprise's shuttlecraft Copernicus was transporting the box to Starbase 25, a group of Kzinti privateers, led by Chuft-Captain and operating from an ostensibly stolen police vessel, the Traitor's Claw, lured the shuttle to a small, ice-bound world in the Beta Lyrae system, using the empty box as bait. In illegal possession of phasers, the privateers ambushed the shuttle's crew of three Starfleet officers, imprisoning them in a police web and confiscating the unopened box in hopes of finding within it a weapon.
The Kzinti indeed found a Slaver weapon among the box's contents, but were initially frustrated and confused by its variable functions. In testing one of its settings, the privateers unwittingly nullified the effect of their own police web, allowing Commander Spock and Lieutenant Sulu to escape with the weapon. Chuft-Captain was wounded by Spock in the process, and thus humiliated.
The Kzinti offered a bargain in which they would trade the life of their remaining prisoner, Lieutenant Uhura, for the return of the weapon, on the condition that Spock allow Chuft-Captain to regain his honor by facing him in personal combat. Although the Starfeet officers refused this offer, the Kzinti nevertheless managed to recapture them and the weapon when the pair inadvertently unlocked the device's most powerful setting and were rendered unconscious by the resulting shockwave.
Ultimately, the privateers were undone in their efforts to use the weapon themselves by its own reasoning intelligence. Determining itself to have fallen into enemy hands, the weapon engaged a self-destruct mechanism, killing the privateers and severely damaging their vessel by generating a disruptor field. (TAS: "The Slaver Weapon")
In 2399, the Kzinti were active near Nepenthe. This was part of the reason that William Riker's home had a security system that allowed them to raise perimeter shields around their residence. (PIC: "Nepenthe")
Kzinti were bipedal digitigrade felines with orange fur, yellow eyes, pronounced fangs, ears resembling bat wings, four-fingered hands, and long tails. Males stood over two meters tall, with broad hunching shoulders and comparatively slender waists and limbs. Their internal anatomy included ribs with vertical bracing and multiple hearts. They breathed atmosphere similar to that of Humans and Vulcans. As carnivores, they were acutely averse to the consumption of plants. Their physical attributes made them powerful combatants; Spock estimated his chances of defeating Chuft-Captain in single combat to be no better than sixteen to one.
Some male Kzinti were capable of reading minds; the effort required was taxing and took time to recover from, often leaving such individuals unhappy and neurotic. There was no sure way to guard one's thoughts from such a telepath, but the experience could be made especially unappealing for the telepath by the subject concentrating intently on images which Kzinti considered distasteful or alien, such as eating a raw vegetable. (TAS: "The Slaver Weapon")
The Kzinti were an aggressive, carnivorous race, who were disdainful of herbivorous species such as Vulcans, and with only marginally more respect for omnivorous ones such as Humans. Kzinti females were considered "dumb animals" without intelligence, and Kzinti males were thus predisposed to underestimate females of other species.
Kzinti placed great importance on individual honor, being obliged to seek personal revenge upon an attacker before calling for help. To be defeated or wounded and left alive was considered by them the ultimate insult, which could only be remedied by single combat to the death.
The Kzinti government was led by the Highest of Kzin. They had police vessels and related equipment for the enforcing of laws, but were disallowed all other weapons by the Treaty of Sirius, yet sought ways of circumventing this and rearming. Other pursuits in which Kzinti engaged included archaeology and astronomy. (TAS: "The Slaver Weapon")
Unlike the other alien antagonists of Star Trek, the Kzinti species was an original creation developed independently of the television production, well before their appearances in Star Trek: The Animated Series, which were in 1973. Science fiction author Larry Niven's Kzinti were first introduced as part of his Known Space universe in "The Warriors", a short story published in 1966. 
After reading Larry Niven's story "The Soft Weapon" and thinking it would be a good fit for Star Trek, D.C. Fontana recommended it to Gene Roddenberry, who agreed. Niven was then invited to adapt his own story into "The Slaver Weapon". According to Fontana, the pink coloration of the Kzinti uniforms and ship in the episode was a result of Director Hal Sutherland being colorblind, and thus unable to discern them as anything but shades of gray. 
In Niven's original work, the term "Kzin" was used to refer to the characters individually in the singular form, while "Kzinti" was used for the plural and adjectival forms. As such, the Elysian councilor was identified by the singular term "Kzin" in the final draft script and storyboards for "The Time Trap". Likewise, "Kzin" was also used in the first and final draft scripts of "The Slaver Weapon". While this reflected Niven's intended usage, the singular term was ultimately never used onscreen because the characters were always referred to in the plural in dialogue.
In a biography of M'Ress, made available through Lincoln Enterprises in 1974, it was stated that Kzinti and Caitians shared common roots, having long ago split and developed into separate civilizations in much the same manner as Vulcans and Romulans. The Worlds of the Federation by Shane Johnson further posited that Caitians were descended from an ancient Kzinti colony.
The Kzinti made no further on-screen appearances in Star Trek after "The Slaver Weapon", but their homeworld of Kzin was included on a star chart created by Mike Okuda that was displayed at Starfleet Headquarters in TNG: "Conspiracy", and which made occasional appearances thereafter in that series, in Kirk's quarters in Star Trek VI: The Undiscovered Country, and in the classroom on Star Trek: Deep Space Nine, though it was never seen closely.
It has often been reported, including by Ian McLean, that the "Cat Dancer" played by Linda Howard (née Fetters) in Star Trek V: The Final Frontier was jokingly referred to by some backstage source or sources in connection with the film as a "Kzinrrett".  In Larry Niven's Known Space works, Kzinrret (plural Kzinrretti) is a term that refers to Kzinti females. Howard has stated that she herself was unaware of being called this. 
Robert Hewitt Wolfe has said that in naming the Tzenkethi, a species first mentioned in DS9: "The Adversary", he may have combined the name "Kzinti" with that of "Tsankth", a deity of piracy in the Glorantha role-playing games. However, Wolfe noted that he could not remember whether he was making an intentional homage to Larry Niven in doing so, and also that he did not imagine the Tzenkethi as felinoid, instead picturing them as "more like the Hakazit [from Jack Chalker's Well World series] ...heavily armored lizard things." 
Jimmy Diggs, who contributed a number of stories to both Star Trek: Deep Space Nine and Star Trek: Voyager (and was a friend of Larry Niven's), was a long-time proponent of the Kzinti returning to Star Trek. He made his first pitch for this in 1994, as an intern on Star Trek: The Next Generation. In the following years, he pitched ideas for Kzinti-involved stories so persistently that Brannon Braga came to refer to them as "Jimmy Diggs' Crazy Cats". 
Jimmy Diggs even penned a screenplay for a proposed feature length CGI-animated film entitled Star Trek: The Lions of the Night, which was to involve Captain Sulu, Uhura, Chekov, and the USS Enterprise facing a Kzinti invasion. For the purposes of pitching this project, Diggs commissioned of diggs3.html concept art by Court Jones, including a design for a Kzinti cruiser called the Dark Stalker. The project had the support of George Takei, Nichelle Nichols, D.C. Fontana, and Majel Barrett-Roddenberry, but ultimately failed to capture the interest of Paramount Pictures. 
Undeterred in his fascination with the Kzinti, Jimmy Diggs later adapted a story idea originated by Neal and Jana Hallford into a teleplay entitled "Kilkenny Cats", which André Bormanis arranged to have Diggs pitch to Manny Coto for the planned fifth season of Star Trek: Enterprise. The story was to involve a Kzinti juvenile being brought on board Enterprise NX-01, and the design of the Dark Stalker was revised by Josh Finney to fit the 22nd century setting of ENT, incorporating elements resembling components of World War II-era sea and aircraft per Diggs' instruction.  Finney created a rough CGI rendering of the vessel for use in pitching the episode. The revised design bears some similarities in its overall arrangement and coloration to 28/ Kzinti spacecraft from the Star Fleet Universe (see "Apocrypha" below).
Jimmy Diggs also created a thirty-five-page "Guide To Using The Kzinti In Star Trek", which he distributed to those whom he described as "key people in the Star Trek and Man-Kzin Wars franchises," including André Bormanis, Brannon Braga, Manny Coto, D.C. Fontana, Larry Niven, Michael Okuda, Jim Baen, and others. This guide included Diggs' ideas of how Kzinti history and culture should fit within the Star Trek universe. 
Following Jimmy Diggs' pitch, Manny Coto promised to fight for the reintroduction of the Kzinti on the show, and was so impressed with a Clint Burgin-sculpted maquette of Diggs' proposed redesign for the species that he kept it in his office. However, the series was ultimately cancelled after its fourth season. 
Diggs subsequently sought to re-adapt "Kilkenny Cats" as an installment of the fan film series Star Trek: Phase II – substituting, by legal necessity, a similar species for the Kzinti – but even that effort was eventually shelved. 
"I sent a fan email to Larry Niven - one of the writers whose work inspired me to want to become a writer, as a kid - and asked his permission to include a reference to his battle-and-honor obsessed Kzinti, who as you know crossed over in one episode of TAS - blowing my ten-year-old, Niven-and-Trek obsessed mind - and also, I've always suspected, helped inspire TNG's reinvention of the Klingons. Mr. Niven very graciously said that would be okay."
Kzinti in novels
Star Trek Log 5, by Alan Dean Foster, begins with a flashback scene of M'Ress' career in Starfleet. Among one of the events is her time as a junior communications officer aboard the USS Hood; the starship was attacked by a Kzin cruiser, killing all of the bridge crew, including the chief engineer, and crippling several systems, including engines and communications. M'Ress, who speaks little Kzinti, volunteers to beam over to the cruiser, which was disabled in a counterattack. The Kzinti are sending a distress call to their homebase, which needs to be cut off, and M'Ress' plan is to recode it to send a distress call from her own crew to a Federation starbase. Her actions result in a promotion to lieutenant after just two years of service and her transfer to the Enterprise. The battle was mentioned in M'Ress' biography by Lincoln Enterprises, where it was stated, "She entered Starfleet just three years ago," placing the event in 2266.
Pocket TOS novel Ishmael, by Barbara Hambly, additionally references the Kzinti. In the book, Maria Kellog, the commanding officer of a particular starbase, is established as having served as chief engineer of the USS Republic. She was one of six Human crewmembers aboard that craft; the rest were Orions, Kzinti, and Trisk. (Chapter 7, Ishmael) Later, Kellogg and Kirk pass two Kzinti in the starbase's corridors, and Kellogg greets them in their native tongue. (Chapter 13, Ishmael)
Kzinti in the Star Fleet Universe
The Kzinti race in the Star Fleet Universe – which has traits setting them apart from the Kzin of Larry Niven's works – has fought wars with all of their neighbors, the Federation, the Klingon Empire and their perennial nemesis, the Lyran Star Empire. They are long-standing allies – or more accurately, co-belligerents – of the Hydran Kingdoms.
The Kzinti Hegemony eventually formed a tentative accord with the Federation, and allied with them in the General War, but they have been involved in major wars with the Klingons and Lyrans, such as the Four Powers War and the General War itself, in which a substantial region of their territory was occupied by their Coalition enemies and two full-scale assaults were made on the Kzinti homeworld of Kzintai.
Eventually, with Federation assistance, they forced the Coalition forces from their territory, but after the war ended, they were involved in a civil war as a disgruntled faction – which had been opposed to the Hegemony's ruling patriarch, sought refuge, developed a power base in the WYN Cluster, and launched an attempted coup of the Hegemony itself in the WYN War of Return.
Also, in a fictional variant of the Star Fleet Universe as represented in the games Star Trek: Starfleet Command II: Empires at War and Star Trek: Starfleet Command - Orion Pirates from Taldren, the Kzinti were renamed and re-interpreted as the more canine Mirak.
The Kzinti also made brief appearances in the 1 March 1982 to 17 July 1982 Star Trek newspaper comic strip The Wristwatch Plantation, by Sharman DiVono and Larry Niven, with art by Ron Harris. The story involved Kzinti starting an invasion in the 2270s. The authors considered publishing the story as a novel or comic book.  However, this did not happen.
The Ferasan playable species in the Klingon Faction in Star Trek Online are a stand in for the Kzinti in that game. Due to legal issues, the developers could not obtain the rights from Larry Niven to use the species in the game. As previously mentioned in the "biography of M'Ress" concerning the Kzinti, background information in the game reveals that the Ferasans and the Caitians were originally one single species. However, violent disagreements about how to employ genetic engineering forced the Caitians to relocate to another planet, Cait. Unlike Kzinti, Ferasan females are fully sentient.