I assume you're referring to Star Trek's Mirror Universe. You managed to watch the two Star Treks that have no Mirror Universe episodes, which is kind of stunning. The Mirror Universe was introduced in The Original Series' (TOS) Mirror, Mirror, but you don't have to watch that episode to get a good introduction to that universe.
The Mirror Universe returned in Deep Space 9's Crossover, which is probably a more palatable introduction to the Mirror Universe if you don't like the 60's style/writing/etc of The Original Series. Just know that Kirk and friends made it over there first, they explicitly reference it in that episode to confirm that they're in the Mirror Universe. Over the course of DS9's run they return to the Mirror Universe periodically.
The last time we saw the Mirror Universe (out of universe chronology), was in an Enterprise (ENT) 2-parter. In a Mirror, Darkly was a 2 part episode that takes place entirely in the Mirror Universe. This arc was a bit more tied to The Original Series, so if you're not a TOS or ENT fan, just skip this.
In this alternate universe, things are more brutal. Familiar characters are more violent and the Federation as we know it doesn't exist. Instead, there is a Terran Empire in the 22nd-23rd century, which has its roots back on Earth, where history is more violent (the Enterprise episodes had an alternate opening along with dialogue that indicated this). Across the galaxy, there are other political differences. In the DS9 episodes, we learn that after the collapse of the Terran Empire, a Klingon-Cardassian Alliance rose to fill the power vacuum. They serve as the main antagonist in the DS9 Mirror Universe episodes.
Memory Alpha provides a handy list of Mirror Universe appearances:
- In a Mirror, Darkly (parts 1 and 2)
- The Original Series
- Mirror, Mirror
- Deep Space 9
- Through the Looking Glass
- Shattered Mirror
- The Emperor's New Cloak