The world of Star Trek is full of alternate universes, different timelines, human-like androids, and aliens with strange powers. It's no surprise that the Captains or their crews will run into people that look identical to the Captains, but couldn't be more different on the inside.
While the trope of an evil twin coming into the picture is far from original, the writers usually managed to create interesting and engaging stories where the evil copy aspect felt natural and even incidental.
Captain Kirk - What Are Little Girls Made Of?
Captain Kirk and crew beam down to a series of caves on an ice-covered planet, attempting to contact a missing exobiologist, Dr. Korby. They discover he is alive, sort of, and he discovered an ancient machine that would create a perfect android copy of anyone, which he uses on Kirk.
An interesting aspect of this episode is how Kirk sabotages his copy in the process of its creation. He doesn't destroy it but makes sure the crew will know it wasn't him, by having negative thoughts about Spock during the copying process.
Captain Georgiou - Mirror Universe
Philippa Georgiou was the commanding officer of the Shenzhou and lost her life in a battle with the Klingon empire. However, when Discovery's crew find themselves in the mirror universe they meet Emperor Georgiou that is sadistic and rules with an iron fist, in contrast to the wise and brave Captain Georgiou.
This is an interesting case because the mirror universe copy has more screentime and development than the original Georgiou. She eventually comes over to the normal Star Trek universe and had significant character development, gradually becoming kinder and more selfless. Her journey to good makes for one of Star Trek's greatest redemptions.
Captain Kirk - Turnabout Intruder
The finale to the original Star Trek series revolved around a former lover kidnapping Captian Kirk and swapping their minds. Janice Lester, in the body of Captain Kirk, finally achieves her goal of becoming a starship captain, something women weren't allowed to do at that time in Starfleet history.
The controversial ending to the original series is an unfortunate example of sexism in early Star Trek. Kirk, while in a woman's body, is inexplicably unable to communicate or help his crew. As for Janice, she is shown to be unable to handle a position of authority. However, the drama is born from the crew dealing with a seemingly mentally unstable Kirk and being forced to mutiny.
Captain Picard - Nemesis
The last of the TNG movies concerns the hostile takeover of the Romulan Empire by Reman separatists, with the ultimate goal of destroying Earth. This deadly plot is put in motion and carried out by Shinzon, a genetic clone of Captain Picard.
This movie is controversial among fans and Nemesis has many parts that don't make sense, but it did introduce audiences to a villain played by a young Tom Hardy. This example also stands out for the Captain copy being played by a different actor. Shinzon has little in common with Picard, except for his staunch willpower, but Picard and Shinzon's interactions are the most engaging parts of the film.
Captain Janeway - Course: Oblivion
The Voyager crew has a lot to celebrate with the wedding of B'elanna to Paris and the fact that their new warp drive should get the home in just two years. However, the warp drive proves to have negative health effects on the crew and the ship itself, turning them into liquid metal.
The Voyager and its crew were all copies in this episode, where conscious liquids gain their memories and believe themselves to be human. These copies aren't evil, but rather have the exact personalities of the crew down to Janeway's unwavering conviction. This makes the episode special because the audience gets to see how Janeway and crew would actually face imminent peril.
Captain Kirk - The Undiscovered Country
The political mystery film that marked the final appearance of the original cast featured a short section with a shapeshifter assuming the form of Captain Kirk. When Kirk and Dr. McCoy are sent to a Klingon prison, a shapeshifting alien helps them escape to the frozen surface of the planet, they are double-crossed and Kirk has to fight himself.
The Kirk prison escape and fight adds some energy to the middle of the film, which breaks up the Enterprise crew's investigation of the apparent firing of a torpedo on the Klingon Chancellor's ship, and his imprisonment by the Klingons is one of the worst things to happen to Captain Kirk. This is also a fun call back to the original series episode "The Enemy Within" when Kirk fights a copy of himself.
Captain Sisko - Mirror Universe
The Captain Sisko audiences normally see is the brave leader of space station Deep Space Nine, but occasionally he or one of his crew members would find themselves in the mirror universe and encounter a very different Benjamin Sisko. Multiple episodes featured this alternate Sisko ending when Captain Sisko had to pose as his deceased counterpart in "Through the Looking Glass."
This copy is much more fleshed out than others, a benefit of the multiple appearances. Mirror Sisko was a slave under the Klingon-Cardassian Alliance, then he became the pirate consort of the evil Kira, and finally made his escape and lead a rebellion against Kira. This also made for some of the best Star Trek mirror universe stories.
Captain Kirk - The Enemy Within
A transporter accident splits Captain Kirk into two different people, one that is authoritative and violent and another that is kind but timid. The "evil" Kirk run amok on the Enterprise, hurting crew members and acquiring a phaser, and the "good" Kirk must find the courage to stop him and reunite the two personalities.
This episode takes the idea of having to do battle with your evil twin and uses it to analyze the psychological duality within a well-balanced leader. Copies of Kirk were far from uncommon in the original series but this episode finally delivered on the promise of Kirk fighting an evil Kirk.
Captain Picard - Allegiance
In his sleep, Captain Picard is abducted by mysterious aliens and imprisoned with three prisoners of varying species. The Enterprise crew are oblivious because a nearly perfect copy of Picard has taken his place, but some strange behaviors eventually raise suspicions among the crew.
This copy is particularly interesting because of the unusual things fans get to see Patrick Stewart do as Picard, that the normal Picard would never do. He leads the crew in song in Ten Forward and even openly pursues Dr. Beverly Crusher, a romance the real Picard never acted on.
Captain Kirk - Mirror Mirror
When the transporter malfunctions, Kirk is transported to another universe where the Federation is an evil Empire and the evil Kirk is transported to the normal Star Trek universe. Can the respective Kirks hide their identity long enough to find a way home?
This classic Star Trek episode introduced the mirror universe, and many of the subsequent series would feature this evil alternate universe. This episode also took the time to make many of the alternate versions of characters visually distinct, like giving Spock a beard and Sulu a scar, a nice touch indicating the very different lives and world they live in.