Corprus Disease is a form of Blight Disease. It drains all of your attributes except Luck and Agility by 1 point and increases your Strength and Endurance by 1 point. During one of my earlier games, I noticed that the disease was progressive. In other words, it drained/added 1 point periodically (about once per day) so I was in pretty sorry shape by the time I got to the Corprusarium. If you have decent skill in Restoration and the appropriate spell, you can replace the drained points each day and keep the increased ones. A 1-point restore spell is pretty darned cheap to cast and has decent chances of success, even at low skill levels. Mind you, everything goes back to normal after you get the cure, but the extra beef is nice while it lasts and your factions will still give you quests, even though their opening dialog makes you think you have cooties or something.
Anyway, Cosades gives you a Dwemer artifact for Divayth Fyr, the Telvanni wizard who runs the Corprusarium. If you are not a member of the Telvanni faction, Divayth Fyr isn't going to be too happy to see you, so you'll need something to suck up with and Cosades has thoughtfully provided it. If you are a member of the Telvanni faction and have a few ranks under your belt, Fyr will be delighted to see you and you can probably keep the artifact for yourself (it's not worth much, so might as well give it to him and save the weight).
Divayth Fyr lives in the Telvanni Wizard Tower called Tel Fyr. It's on an island southwest of Sadrith Mora, so you'll either need to be able to fly, water walk or swim. You can get to Sadrith Mora by boat from Ebonheart, Tel Branora, Vos or Tel Mora. Otherwise, use the Guild Guide at the Mages Guild in Balmora and just teleport to Wolverine Hall. A walking excursion from Balmora to Tel Fyr should take you a day or two and it isn't too difficult if you can handle the wilderness encounters.
A few words of caution before you even enter Tel Fyr. First, you'll need to be able to levitate in order to get to Divayth Fyr's chambers (high Acrobatics will work, too). This is a good thing to keep in mind when dealing with most of the Telvanni wizards. Second, no matter what they do or what they say, do not kill anyone. Killing one of the Corprusarium inmates will drop Fyr's disposition to abysmal levels and it will take some serious sucking up to get it back up to where you can finish this quest. If you are not Telvanni, you may not be able to bring it back up. Kill one of his daughters, the Warden or Yagrum Bagarn and you're completely hozed. He will not give you the cure under any circumstances. Although the dialogue indicates that the cure is a potion, it is actually a spell. So don't go thinking that you can just kill him and take it off of his body. Keep this fellow happy.
When you enter Tel Fyr, take the first right to get up to Fyr's chambers and be able to levitate. Fyr (yes, that's a set of Daedric armor he's wearing) will ask for the artifact as soon as you open the dialogue. Since a Dwemer anything is heavy, just give it to him. Talk to him about anything that's available in his dialogue threads to learn about corprus disease, about the corprusarium, about a possible cure, whatever. Fyr sends you down to the Corprusarium so that you can get a look at what's in store for you if you don't take his cure and while you're there, pick up a pair of boots that one of the inmates is repairing for him.
So drop back down to the main floor and head back for the entrance. Keep going down until you get to a room on your right with one of his daughters in it. There is a pillar in that room that blocks your view of the door to the Corprusarium. Go through that door and you'll find yourself in the Corprusarium. The Warden (an Argonian) should be directly in front of you next to a gate. Go through the gate to enter the Corprusarium proper.
You'll find Corprus Stalkers and Lame Corpruses galore wandering around down here. All of them will most likely try to attack you. Just run away (a "Calm Creature" effect might work - I've not been successful with it , but your mileage may vary). Straight ahead would be good since there is a door there that will take you into the Corprusarium Bowels.
Once you're into the Bowels, just keep bearing left until you see a couple of NPCs back against the rear of one of the chambers. One of them is one of Divayth Fyr's daughters and the other is Yagrum Bagarn. If you happen to have a Guarskin drum (you might have picked one up in your travels or there is one in the outer portion of the Corprusarium), you can give it to the girl and she will calm the inmates so that they don't attack you. If not, just talk fast.
You may have run into Yagrum Bagarn as part of a couple of other quests. He's quite a character and a good source of information on the Dwarves. Be sure to pick up the boots that Fyr wants, but feel free to talk to him about whatever is available in his dialogue options. When you're done, head back the way you came and report to Fyr.
It would not be a good idea to kill Yagrum Bagarn, even after completing this quest. If you are so bloodthirsty that you would even consider it, you will probably screw yourself somewhere down the road by killing someone who has some vital information and Yagrum is the only other source of that information.
Return to Fyr with the boots and select his "potion" dialogue option. You must agree to drinking it there (you can't do otherwise). You'll get some additional dialogue from Fyr, but the important part is that the negative effects of the corprus disease have been removed and you are now immune to Common and Blight diseases.
As for what else is in the Corprusarium, the chest next to Yagrum and Uupse Fyr contains the artifact Volendrang and the closet behind them contains a full set of Dwarven armor and quite a few skill books. There is a book describing most of the artifacts in the game. There is also a Dead Adventurer laying around.
There isn't much in Onyx Hall, but there is quite a bit of stuff laying around Fyr's tower. There is a closet with the Savior's Cuirass, books galore (none of them skill-raising), potions and ingredients, and Fyr himself has a set of Daedric armor (no Helm or Shield) and a Daedric dagger. Wait until after you're cured before trying to steal any of it.
Report back to Cosades for the next little job.