In TOS episode Mirror Mirror, Kirk, McCoy, Scotty and Uhura beam back from a planet during an ion storm which causes a Transporter accident. It swaps them with their counter-parts from the Mirror Universe. The Mirror Universe is exactly the same except the Federation is a Nazi-like evil empire. Spock has an evil gotee, the officers wear more flamboyant uniforms and their logo is the earth with a sword through it.
The Vulcans are sort of slaves to the human-centric Federation though Spock is still First Officer. Instead of joining with other races, the "Terran Empire" conquered them. The Mirror Universe used devices called agonizers to instil pain as punishment. I think thats what people think the scar on Lorca's back is, because its a triangular device and I believe his scar was too.
To move up the ranks in the Mirror Universe, you basically over-throw your commander. Kirk had over-thrown Pike to become Captain of the Enterprise, for example.
Mirror Universe Spock was still "logical" and stated that the dictatorial ways of the Terran Empire was unsustainable and would collapse eventually. When Prime Kirk was still in Mirror Universe, he implored Mirror Spock to be a leader for change.
That was the last we saw of Mirror Universe until DS9. We then learned that Mirror Spock had taken Prime Kirk's words to heart and rose up to become a leader in the Terran Empire and proposed many reforms including disarmament. Once disarmed, all the enemies the Terran's had made joined forces against them, especially the Klingon-Cardassion Alliance and enslaved Terrans and Vulcans.
Enterprise also visited the Mirror Universe as a sequel to the TOS Episode The Tholian Web. In the TOS episode, the USS Defiant (not the DS9 Version, but a Constitution-Class ship) sent a distress call. The Enterprise went after it and found the Defiant stuck in an interphase, phasing in and out of the Prime Universe.
In the time of Enterprise but in the Mirror Universe, we learn that the later Interphase was caused by the Tholians of Enterprise time creating a warp bubble or something and luring the Defiant from the Prime Universe to capture future technology. Archer's Mirror Enterprise intervenes (Captained by Forrest instead of Archer) and captures the Defiant. It was a neat episode because the Defiant was identical to TOS Enterprise so we got scenes of Archer et al on that ship and in the TOS uniforms.
The Mirror Enterprise episode ended with Mirror Yoshi Sato having killed her way up the chain of command and was in control of the Defiant. The series was cancelled before they could re-visit it.
The cool think about the Enterprise episode was it was entirely in the Mirror Universe, there was no event that crossed us over. Even the opening credits were changed to one more fitting the Terran Empire.
Its never been said what split the Mirror Universe from the Prime but speculation is either the Nazi's winning the war (which doesnt make a ton of sense but it could have been a precursor). Many fans like to believe it was Kirk saving Edith Keeler in the Guardian of Forever episode.
The original idea for an Enterprise Mirror episode resolved this issue by showing the creation. Originally, the concept was based on a Mirror Universe weapon called the Tantalus Field. In TOS Mirror episode the Tantalus field was used to monitor anyone and, with a push of a button, disintegrate them. It was plundered from an alien world by Kirk, giving him his great power in the Mirror Universe.
William Shatner actually proposed the idea to the Enterprise team suggesting the Tantalus Device didnt disintegrate people, it sent them back in time to a penal colony. The pitch was Archer's Enterprise would stumble across this penal colony and find Mirror Kirk (Tiberius) there. Mirror Kirk would try to use the Enterprise' transporters to return to his Universe but finds it doesnt exist yet. Mirror Kirk and Prime Archer work together to determine the divergent point and in effect, creates the Mirror Universe for Tiberius to return to.
That episode never happened because Berman counter-pitched Shatner that he instead play "chef". Shatner thought it was awful.
The other cool thing about the Enterprise episode was it opened with the Vulcan landing scene from First Contact including using actual footage from the movie. But at the moment Cochrane tries to give the Vulcan salute, he pulls out a phaser rifle and shoots the Vulcan.
I think the only reason the Mirror Universe is speculated about Discovery so much is Jonathon Frakes "slipped" up and said there was a Mirror Universe episode. Some people think Discovery takes place in the Mirror Universe (it doesnt) to explain the differences or takes places a third universe and that Discovery will eventually return to the Prime universe which will look closer to what we know of TOS.