I see many mentions to another universe in Star Trek questions, but I've never actually remember seeing an episode of Star Trek that dealt with this other universe. I've watched all of TNG and all of VOY and nothing in there seems to match up with whats being referenced.
What episodes do I need to watch to familiarize myself with this plot element?
The mirror Universe is a parallel reality wherein Earth is far more militaristic than it is in the "prime" Universe, and wound up conquering its way to galactic prominence to form the Terran Empire. Memory Alpha has a whole article devoted to the details.
It was introduced in the original series' episode, "Mirror, Mirror" and became a prominent side-arc in Deep Space Nine. It was not featured at all in The Next Generation or Voyager, which is why you haven't run into it. It was, however, featured towards the end of Enterprise for a two-part episode.
If you're just interested in watching the mirror universe episodes, here's a chronological list (also from the above Memory Alpha article):
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:
You'd need to watch "Mirror Mirror" (as mentioned by Donald.McLean) from TOS and "Parallels" from TNG.
Here's a link to the wikipedia page for "Parallels." http://en.wikipedia.org/wiki/Parallels_(Star_Trek:_The_Next_Generation)
The 2009 movie is based on the concept of parallel realities, not necessarily universes, wherein minor changes in a timeline can produce entire new realities. See our discussion here. Do parallel realities take precedence over time travel in the Star Trek Universe?
Plus, if it split off, Zephran would have been carrying a gun in the normal universe seconds before the split. It is possible that he considered attacking Geordi La Forge, but it shows no sign of that in the movie. My theory is the world split sometime in the 1950's, which would explain why the first moon landing has the Terran flag, not the American, and why the astronaut has an EV Suit. Theoretically,the first moon landing happened just a few years before first contact. The early split off would suggest that the Empire never encountered the Borg, so they never had to go back and stop them.
