This is a list of the Klingon-centric episodes and movies from throughout the Star Trek franchise:
Star Trek: The Original Series
1x26 "Errand of Mercy"
The very first appearance of Klingons, featuring the first appearance of Kor, and introducing the ongoing conflict between the Klingon Empire and the United Federation of Planets, something the Organians attempt to solve.
2x11 "Friday's Child"
The first indication that the Organian solution didn't stick.
2x15 "The Trouble with Tribbles"
Perhaps the most famous Klingon episode ever, featuring the unlikeliest element of little furballs who prove to be their true archenemy, plus a double agent and a warrior who incites Scotty by claiming the Enterprise should be "hauled away as garbage." Also features the first appearance of Koloth.
2x19 "A Private Little War"
The rivalry continues.
3x7 "Day of the Dove"
A rare moment where Starfleet and Klingons can agree on something is when they discover a common enemy. This features the first appearance of Kang.
3x13 "Elaan of Troyius"
The writers used both the Romulans and Klingons as allegories of the Cold War. This is very much one of the episode reminiscent of the Vietnam quagmire that was a part of that.
3x22 "The Savage Curtain"
The final appearance of the Klingons in the series, this also features the debut of the legendary Kahless, who gave birth to the warrior culture of the Empire.
Star Trek: The Animated Series
1x5 "More Tribbles, More Troubles"
Not the last sequel to the classic original Tribble episode but certainly the least known, this one also features another appearance by Koloth.
1x12 "The Time Trap"
Featuring another appearance by Kor.
Star Trek: The Motion Picture
Featuring the debut of the ridged foreheads and thus our modern conception of Klingons.
Star Trek III: The Search for Spock
A Klingon played by Christopher Lloyd attempts to use the Genesis Planet created in the previous movie as a weapon, but ends up with little cooperation. As a consolation, he has Kirk's son David Marcus murdered and the original Enterprise is destroyed. This counts as the biggest victory the Empire ever scores against Starfleet and the Federation.
Star Trek IV: The Voyage Home
A Klingon ambassador won't rest until Kirk is properly punished, which to Starfleet's mind is only relevant in that he disobeyed a direct order not to undertake his rescue of Spock. This movie leads to the big dramatic political ideas that begin to dominate Klingon affairs in subsequent appearances.
Star Trek V: The Final Frontier
This movie seems to be an aberration in just about every regard. The Klingons here end up become very nearly friends with Kirk's crew, and he himself admits that the exploits of the general they encounter were required reading at Starfleet Academy.
Star Trek VI: The Undiscovered Country
The Empire undergoes a crisis that leads some of its leaders to admit that it can no longer afford an ongoing rivalry with the Federation. Kirk is framed for the chancellor's assassination and a conspiracy that reaches across loyalties is exposed. This is the most Klingon of any Star Trek movie to date, required viewing for any fan. The joke that Klingons wrote Hamlet became an actual stage translation, much as the Klingon language had undergone serious study. There's a dictionary and everything!
Star Trek: The Next Generation
1x1/1x2 "Encounter at Farpoint"
The first appearance of Worf, who is a Klingon serving in Starfleet.
1x20 "Heart of Glory"
The first episode of the series to feature Klingons other than Worf.
2x8 "A Matter of Honor"
Riker participates in an exchange program that has him serve temporarily as first officer aboard a Klingon ship.
2x20 "The Emissary"
First appearance of K'Ehleyr, Worf's ideal Klingon match.
3x15 "Yesterday's Enterprise
A unique look at the lost era between The Original Series and The Next Generation, a period that saw great conflict between the Klingons and Romulans, which led directly to the death of Worf's parents.
3x17 "Sins of the Father"
Another classic episode that reveals more than you ever wanted to know about Klingons politics that wasn't covered in The Undiscovered Country, which happens to bring to an end everything left of Worf's connection to his own people. The episode, incidentally, also introduces his brother Kurn and the great rival Duras.
K'Ehleyr and Duras return, and neither survives the episode. This is a big Klingon episode and an important Worf episode as well. Also features the debut of Gowron, plus Alexander, Worf's son.
4x21 "The Drumhead"
Really has very little to do with Klingons, but I'd recommend this one anyway.
4x24 "The Mind's Eye"
Years before the Dominion attempted the same thing, the Romulans try to manipulate a wide-scale conflict between Klingons and the Federation.
4x26 "Redemption, Part 1"
Civil War, Klingon style! Features Gowron plus Lursa and B'Etor, the infamous sisters of Duras. Worf temporarily walks away from Starfleet in order to join his brother Kurn in the conflict.
5x1 "Redemption, Part 2"
Conclusion of all that, plus the appearance of a Romulan who looks strangely like someone last seen in "Yesterday's Enterprise," Tasha Yar, a series regular in the first season and the original holder of the security post Worf enjoys.
5x10 "New Ground"
Worf gains full-time custody of Alexander.
A horrifying look at the furthest extent of Klingon honor.
6x16/6x17 "Birthright, Parts 1 and 2"
Worf explores an unexpected legacy of the past conflicts between Klingons and Romulans.
6x23 "Rightful Heir"
Worf supports the claims of the apparently returned Kahless. Gowron does not.
The final series statement on Worf and his son Alexander, who struggles with his Klingon nature. Also features the Duras sisters.
7x25/7x26 "All Good Things..."
In the final episode of the series, an alternate future sees renewed hostilities between Klingons and the Federation.
Star Trek Generations
It's the end of the Duras sisters!
Star Trek: Deep Space Nine
1x3 "Past Prologue"
Chronologically, the sisters do have this appearance as well.
1x18 "Dramatis Personae"
Klingons feature in this episode, for the record.
Kor, Kang and Koloth from The Original Series return in a big way, old allies of the Dax symbiont who recruit new host Jadzia to complete one last great adventure.
3x3 "The House of Quark"
The second most unlikely Klingon episode ever of course features a Ferengi, as Quark stumbles into responsibility for a whole house, and the bride who goes along with it.
Klingons run amok in this episode that is otherwise not really about them
4x1/4x2 "The Way of the Warrior"
Worf becomes a series regular as of this epic episode that also brings back Gowron and introduces Martok. This is the start of a protracted conflict instigated by the Dominion.
4x9 "The Sword of Kahless"
Kor returns as he and Worf struggle over the eponymous relic.
4x14 "Return to Grace"
Cardassian foil Dukat undertakes a personal war against Klingons.
4x15 "Sons of Mogh"
Worf's brother Kurn returns, but the conflict between them is much the same as before.
4x18 "Rules of Engagement"
As is typical for Worf, he's caught between Starfleet and Klingon worlds in this episode.
4x20 "Shattered Mirror"
The Deep Space Nine continuation of the Mirror Universe saga first featured in the Original Series episode "Mirror, Mirror" featured Klingons prominently, but this time with Alternate Worf involved, it's much more obvious.
5x1 "Apocalypse Rising"
Everyone figures out that the Dominion was manipulating the Klingons into war when Martok is exposed as a changeling infiltrator.
5x3 "Looking for par'Mach in All the Wrong Places"
Worf meets Quark's Klingon bride, but ends up realizing his true match is Jadzia.
5x4 "Nor the Battle to the Strong"
The first great war episode of the series involves Klingons, not the Dominion.
5x6 "Trials and Tribble-ations"
Famusly integrates footage from "The Trouble with Tribbles" into a new, time-traveling adventure.
5x14/5x15 "In Purgatory's Shadow"/"By Inferno's Light"
Worf battles Jem'Hadar in a Dominion prison camp, where incidentally he meets the real Martok. Gowron, meanwhile, must be persuaded to end the conflict Klingons and the Federation now that the Cardassians have joined the Dominion in anticipation of a much worse war.
5x21 "Soldiers of the Empire"
Another episode set aboard a Klingon ship, with Martok once again along for the ride.
6x3 "Sons and Daughters"
Worf's son Alexander returns. Though he's phsyically much more mature, he's still not much more of a typical warrior.
6x5 "Favor the Bold"
As the initial onslaught of the Dominion War draws to a close, it's nice to note that the Klingons are indeed fighting alongside Starfleet.
6x7 "You Are Cordially Invited"
Witness the rigors of a Klingon wedding as Worf marries Jadzia!
6x26 "Tears of the Prophet"
And, uh, Jadzia dies this episode.
7x7 "Once More Upon the Breach"
Kor returns for one last inglorious hurrah, with Worf attempting to mediate between the old warrior and Martok.
7x22 "Tacking Into the Wind"
As the series and the war draw to a close, it becomes increasingly clear that Gowron doesn't have what it takes to see the Empire reach the finish. So it's Worf again who helps change the course by, um, helping Martok become the new chancellor.
7x25/7x26 "What You Leave Behind"
Martok does indeed help the Klingons and their allies (i.e. the Federation) win the war!
Star Trek: Voyager
Introduction of B'Elanna Torres, who like K'Ehleyr before her is half Klingon and half human.
A bold look at the conflict between B'Elanna's dueling halves as they're literally split apart.
Features an appearance by Kor!
3x22 "Real Life"
The Doctor's short-lived holographic family features a rebellious son with Klingon friends. It only figures.
4x3 "Day of Honor"
B'Elanna glumly celebrates a Klingon holiday that helped served as the basis for a special publishing event for the Pocket Books novels.
6x3 "Barge of the Dead"
B'Elanna visits the Klingon afterlife. She has all the fun!
B'Elanna worries about the genetics of her pending child with Tom Paris.
A generational ship of Klingons!
Future Janeway steals time-travel technology from the Klingons. Of course!
Star Trek: Enterprise
1x1/1x2 "Broken Bow"
The mission begins with a Klingon crisis. Of course!
1x14 "Sleeping Dogs"
Archer rescues more Klingons.
Klingons Original Series style!
Klingons Undiscovered Country style! Plus an ancestor of Duras also named...Duras.
Doesn't feature Klingon per say, but does continue Archer's debacle involving Klingons.
2x26 "The Expanse"
The Xindi arc of the following seasons begins with the end of the Ancestor Duras arc.
The origin of the smooth forehead begins with the start of the Augments arc.
The origin of the smooth-headed Klingons! And the final Klingon-centric episodes!
Klingons were cut from the reboot. True story. That's what Nero was up to after making George Kirk a martyr and before destroying Vulcan. He was a prisoner.
Star Trek Into Darkness
Klingons feature in this one. For real.