I've been wondering how to best cover this flashpoint for a while, because I worry that a lot of what I'm going to say is basically just going to be a copy and paste of my post about Assault on Tython. The two flashpoints are just that similar! I haven't come up with a solution, but I figured I should just get it over with.
Korriban Incursion was added to the game in March 2014, as part of patch 2.7, "Invasion". Together with Assault on Tython, which was released at the same time, it formed the first part of the Forged Alliances story arc, originally designed for level 55 characters (the level cap at the time) and now labelled as "Shadow of Revan: Prelude" in game. This arc would eventually cover four different flashpoints which initially didn't have a solo mode, forcing you to group for every single step along the way.
Fortunately queue pops were quick however, as it was released as a tactical (role-neutral, though not level neutral at the time) version only. Solo and hard mode weren't added until 3.0, and since the 4.0 revamp the tactical version has been available from level 15 and the hardmode from 50 onwards.
Korriban Incursion takes place on Korriban as we know and love it, only it's being invaded by the Republic. As a Republic player you play through the invasion yourself, while as an Imperial player you come in later and re-claim the Sith Academy from the invaders.
Depending on your faction, you fight either Sith instructors and acolytes training at the Academy, or Jedi and their padawans, whom they decided to bring along to an important invasion en masse for some reason (a problem that Assault on Tython has as well and a side effect of all the mobs just being renamed reskins for the faction that is taking the planet back).
While on Tython the Empire gets to meet the more interesting characters as the original invader, in Korriban Incursion it's the other way round. The first boss for Republic players is Lord Renning, known to Sith players as a quest giver who is obsessed with Tuk'atas and accordingly sics them on you during this fight as well. (For comparison, the random Jedi doing the same against Imperial players feels rather weird.)
Inside the Academy, you end up taking out Inquisitor Arzanon, Lord Solence, Lord Cestus and Overseer Ragate - all four of them characters that are also present in the Sith starting experience, with two of them acting as quest givers. Again, Imperial players just fight some random invading Jedi in their stead.
Finally, Republic players seal their victory by defeating Darth Soverus - who is officially a Dark Council member but had never been heard of before this flashpoint, which means that his death is not all that meaningful. His main ability is that he summons a probe droid add that stuns a random party member until it's killed - though on tactical mode the probe eventually expires on its own even if nobody attacks it, which has saved many a pug from wiping. (On solo mode it just appears and explodes, simply requiring you to dodge a red circle.)
When Imperial players retake the planet, they are up against all the random Jedi I already mentioned, as well as Commander Jensyn, who is guarding the Sith Academy. On Republic side he was involved with the assault and showed up at the very end, so apparently he's just been sitting in the ruined Dark Council chamber the entire time since then. He once again has the same abilities as Darth Soverus.
Other fights worth mentioning are the assault droid of many coloured circles, who is once again identical for both factions except that he's got a different designation, and the bonus boss on hardmode, a Sith or Jedi who is actually pretty damn hard as he has an AoE Force choke that also does massive damage, meaning that your healer needs to cast some proactive heals at the right time (e.g. a big group heal over time) or your whole group will die as soon as everyone gets stunned.
The Republic has decided to launch a daring smash-and-grab attack on Korriban, directed by a certain Colonel Darok and SIS agent Theron Shan (newly introduced to the game, only mentioned in the novel Annihilation until then). You get to be the (wo)man on the ground and land in the lower valley in front of the Sith Academy. On your way in you run into some captured slaves, whom you free - you can then either insist on finding the resources to get them out or make them tag along with primitive weapons to assist you with the fighting (which usually means that they die on the next boss).
Inside the Academy you fight more and more powerful Sith until you subdue the lone Dark Council member that's present. When Commander Jensyn shows up, the Darth mutters some strange things, expressing surprise but also sounding like he actually expected Jensyn somehow, and the Jedi stabs him. You can express annoyance about this, seeing how Soverus was already defeated anyway, but it makes no difference.
When you come back to retake the Academy as an Imperial player, you land in the same spot in the valley. Where the Republic found slaves, you find some weak acolytes that have been imprisoned, and once again you can simply let them go or draft them to (feebly) fight for you. Inside the Academy you meet Commander Jensyn (who's apparently been hanging out in the Dark Council chamber the entire time) and kill him. In his death throes he also mutters something cryptic about how "it's happening" and we can't stop it.
Like Assault on Tython, I quite like Korriban Incursion. Its story is well presented and I didn't mind the original grouping requirement, even if it wasn't everyone's cup of tea. It's fun to see a familiar location in a different context, something that Bioware hasn't repeated so far, with all other flashpoints taking place in unique environments. I originally felt so-so about the tactical thing, but with everything that's been changed in 4.0, Korriban Incursion is one of the few flashpoints that doesn't feel overtuned even if you end up with a weaker group, which is nice. And of course the introduction of Lana and Theron has turned out to be quite important in the long run...