This is not a quest, but it is something that happens to you after you start actively pursuing the Main Quest. Specifically, it will start when you have accepted Antabolis' mission to retrieve the Puzzle Box and continues throughout the Main Quest, unless you do something about it.
The first clue that something is not-quite-right will be otherwise normal folks who suddenly start talking to you about Dagoth Ur and the Sixth House. The conversation should show up automatically when you get within a certain distance of the person, but you may have to initiate the conversation.
If you pursue the conversation thread about the "Sixth House," the person's disposition will drop by 10 points each time. When it reaches zero (a little higher a bit later in the game), they will attack you. Now these are just ordinary NPCs who happen to be getting bad vibes from Dagoth Ur's folks, so killing them isn't going to get you anything. But if you provoke them into a fight, you'll have to kill them.
I had one fellow in Balmora who insisted on shooting at me with his bow (he wasn't very good with it) for several days. When he ran out of arrows, he decided to beat me to death with his fists. I knocked him unconscious a few times and went about my business, but he still persisted. I finally wound up killing him just to get a little peace and quiet.
The second thing you will notice is that you're going to start having bad dreams if you sleep in Balmora, Ald'ruhn, or Vivec. In and of themselves, the dreams aren't any great shakes. You say "rest", you get a dream, you say "rest" again and get your beauty sleep.
You can get four dreams, each triggered by one of the early Main Quest quests. The first dream can occur when you have done the Puzzle Box quest. The second dream can occur when you have done the Addhiranirr portion of the Mission to Vivec quest. The third dream can occur when you have done the Mehra Milo portion of the Mission to Vivec quest. And the fourth dream can occur when you have done the Hassour Zainsubani quest.
Each of the dreams tells a little bit of a story, with each building on the one before. When you finally have the fourth dream, you will find that the figure in the golden mask in each of them is Dagoth Ur and he addresses you as Lord Nerevar Indoril reborn.
The third thing you should notice (it doesn't always happen for me) is that following each of these dreams someone attacks you, usually from behind, right after you wake up. My solution to the problem was to just make sure there was no room behind me when I clicked on the bed. Problem solved.
You can follow up on some of this with Caius Cosades. Initially he thinks you're going loopy on him, but finally he suggests that sleeping outside of town might be the wiser course until you can figure out what's going on.
The last thing you should notice is the appearance of people in the wilderness who show as "Dreamer" when you get close enough. Unless you initiate a conversation with them, they will remain pretty much harmless.
OK. What to do about all of this? Eventually you're going to figure out (or someone will tell you) that you have to go destroy the Sixth House bases to make it stop. There are four of them that will take care of the Sleepers and Dreamers:
Here's the crux of the problem, though. Simply killing Dagoth Ur and completing the Main Quest will not make these folks go away. You must take out these specific bases. On the bright side, though, you've got some serious reputation bonuses piling up. After taking care of the bases, talk to the following people and you should see a 1-point boost to your overall rep after talking to each one (that's 14 points total):