Star Trek: The Original Series first introduced the Mirror Universe in "Mirror, Mirror," the only episode of the series to deal directly with the idea of a darker parallel universe. However, this idea was so interesting that it subsequently spread to other shows as the franchise grew.
The Mirror Universe is a parallel universe that exists in the same space as the Prime Universe, only on another dimensional plane. The same characters, places, and even starships exist in the Mirror Universe, but everything there is the antithesis of what it is in the Prime Universe, meaning that characters who are good in the Prime Universe are portrayed as evil, and humane organizations like the Federation are instead ruthless genocidal empires.
Unsurprisingly, the idea of a parallel universe where the audience is able to see characters show a much darker side of themselves ended up being fertile territory for Star Trek. To date, there have been 14 episodes across 4 series that are set in the Mirror Universe. This has made for a rich and complex history, one that contains many callbacks to other episodes and is fascinating to explore.
"Mirror, Mirror"
A transporter accident causes Captain Kirk, Doctor McCoy, and Lieutenants Scotty and Uhura to switch places with their Mirror Universe counterparts. The group attempts to find a way back to their universe while pretending to be their counterparts. Along the way, they learn about the Terran Empire, the ruthless version of the Federation where in-fighting, mutiny, and assassination are common occurrences. Ultimately, they are able to send themselves back with a little help from Mirror Universe Spock. Before leaving, Prime Kirk and Mirror Spock have a discussion, in which Spock reveals that he believes the Empire cannot sustain itself. Kirk attempts to convince Spock to push for reform within the Empire, and Spock seems intrigued by the idea, telling Kirk he will consider it.
"The Tholian Web"
While it is true that "Mirror, Mirror" is the only Original Series episode that deals directly with the Mirror Universe, "The Tholian Web" establishes canon that becomes relevant to the Mirror Universe in other series. The episode finds the Enterprise's crew on a mission to figure out what has happened to the USS Defiant. They manage to locate the Defiant, only to find out that the ship has been trapped in an area of spatial interphase, where space around the ship is breaking apart, causing the crew to go mad and murder each other. While attempting to figure out a way to rescue the ship, the Enterprise is attacked by a race called the Tholians, and the ensuing battle causes a rip in the spatial interphase, pushing the Defiant out of their universe and stranding Captain Kirk in the space between universes. The Enterprise crew rescues Kirk from interphase, but the true fate of the Defiant is not revealed by the end of the episode.
"Crossover"
The next time the Mirror Universe shows up is in Doctor Bashir manage to escape with help from the counterparts of Benjamin Sisko and Miles O'Brien, nicknamed "Smiley" by Sisko. Along the way, Sisko begins a Terran rebellion against the Alliance.
"Through the Looking Glass"
Captain Sisko is abducted by Miles "Smiley" O'Brien and taken to the Mirror Universe using a multidimensional transporter. Smiley tells Sisko that his counterpart has been killed recently, and asks Sisko to assume his counterpart's role so that he might convince his wife, Jennifer Sisko, to defect from the Alliance where she has been working as a scientist fighting against the Terran Rebellion. In the Prime Universe, Jennifer Sisko is dead, and fear of seeing her counterpart die at the hands of the rebels propels Sisko into agreeing to help. He and a group of rebels travel to Terok Nor and manage to convince Jennifer to them, effectively poaching her from the Alliance and dealing them a crippling blow. Jennifer and Captain Sisko reconcile before he leaves to return to his own universe.
"Shattered Mirror"
Mirror Jennifer Sisko adducts Jake Sisko, taking him to the Mirror Universe in an effort to get Captain Sisko to come and help the rebels build a version of Deep Space Nine's ship, the Defiant (named after the Original Series ship). Sisko reluctantly agrees to help and ends up aiding in a battle between the rebels and the Alliance, which the rebels win, continuing their trajectory towards victory. Jennifer gets to know Jake Sisko, and the episode raises the possibility of the three of them becoming a family again, with Jennifer replacing her Prime Universe counterpart. These hopes are dashed when Jennifer is killed by Intendant Kira during her escape from Terok Nor, leaving the Siskos to mourn her ing and return to their own universe without her.
"Resurrection"
The Mirror version of Bareil Antos escapes to the Prime Universe ostensively seeking asylum, and despite the fact that he is very different from the Vedek Bareil Major Kira was in love with, the two begin a relationship not long after his arrival. It is later revealed however that Bareil is only in the Prime Universe as part of a plot with Intendant Kira to steal one of the scared Bajoran Orbs. Ultimately, Bareil is conflicted about stealing the Orb, and turns against the Intendant at the last minute, stunning her and agreeing to take her back to the Mirror Universe. Kira urges him to stay, but Bareil does not believe he is worthy of a life with her and leaves Kira heartbroken.
"The Emperor's New Cloak"
Grand Nagus Zek is captured by the Alliance while on a trip to the Mirror Universe and held ransom by Regent Worf. The Regent demands the Quark and Rom provide him with a Klingon cloaking device from the Prime Universe, and while delivering it, the two Ferengi are also captured and forced to help Worf install the device on his ship. They instead help the rebels take over the ship and steal back the cloaking device, dealing another crippling blow to the Alliance and all but ensuring the rebellion's victory in the near future. "The Emperor's New Cloak" marks the final Mirror Universe episode for Star Trek: Deep Space Nine.
"In a Mirror, Darkly Part I"
the Enterprise crew are successful in absconding with the Defiant by the end of the episode after the Enterprise is destroyed by the Tholians.
"In a Mirror, Darkly Part II"
With the remaining crew of the Enterprise now on board the Defiant, Commander Archer attempts to use the ship in his plan to grab power, his ultimate goal to overthrow the Emperor and take control of the Empire and Starfleet. Along the way, he thwarts an attempted coup led by T'Pol, but never manages to become Emperor, ultimately double-crossed by his lover, Hoshi Sato. Sato poisons Archer, takes control of the Defiant, and declares herself Empress. Although the rest of her rise to power is not shown, the episode makes it clear that she will be successful.
"Into the Forest I Go"
Lieutenant Paul Stamets trapped in the mycelial network. While the episode only depicts a few minutes of Discovery's initial arrival, it is the catalyst for the rest of the storytelling that goes on in the Mirror Universe during the season.
"Despite Yourself"
Salvaging a data core from a damaged ship, Discovery's crew learns where they are and hatches a plan to get home, hoping that information about how the Defiant crossed over will help them return. Captain Lorca, Ash Tyler, and Michael Burnham infiltrate the ISS Shenzhou, where the Mirror Michael was Captain, posing as their Terran counterparts to retrieve data about the Defiant. Michael and Tyler's relationship deepens as they turn to each other for solace, even as Tyler struggles with the personality of Voq implanted in him by the Klingons.
"The Wolf Inside"
Michael Burnham struggles with maintaining her cover as her Mirror Universe counterpart as she attempts to find a way to save a colony of non-Terran rebels even as the Empire has ordered the Shenzhou to destroy them. Michael also uncovers Tyler's Klingon personality. Voq takes Tyler over and tries to kill her, resulting in Michael having to fake his execution for attacking her while in reality having him beamed back to Discovery with the data they need on the Defiant. At the end of the episode, Emperor Philippa Georgiou arrives in the Terran flagship and demands to see Michael and Lorca.
"Vaulting Ambition"
With the help of his Mirror counterpart, Lieutenant Stamets is able to find his way out of the mycelial network and warns Discovery that Emperor Georgiou is using a biological spore weapon developed by his counterpart to corrupt the network. Meanwhile, Michael attempts to win Georgiou to her side after revealing that she is not her Mirror counterpart. Georgiou in turn reveals that Captain Lorca has actually been in his Mirror counterpart all along, assuming Prime Lorca's role after he accidentally crossed over to the Prime Universe. After breaking free from his torturers on the Charon, Lorca gathers followers and restarts his coup against Georgiou.
"What's Past Is Prologue"
Michael and the Emperor are ultimately successful in thwarting Lorca's coup and killing him once and for all. Discovery then destroys the Charon so that the biological spore weapon does not continue corrupting the mycelial network, and Stamets is able to pilot Discovery back to the Prime Universe. Michael impulsively rescues Georgiou before the Charon is destroyed, bringing her back to Discovery where she accompanies the crew back to the Prime Universe.
While "What's Past Is Prologue" is the final episode of Star Trek set in the Mirror Universe to date, Discovery has continued to have a connection to it through the character of Georgiou. Discovery season 3 has also added a bit more canon to the Mirror Universe, with Georgiou learning that the distance between the two universes had been expanding, and as of 3189, there hasn't been a crossing in over 500 years. If Star Trek decides to set any more episodes in the Mirror Universe in the future, this fact would be an interesting storyline to explore.