Corprus, meaning skin blight in Ald Chimeris, is in Morrowind a name used for any of a number of ailments affecting the skin. During the Third Era, it became especially associated with the so-called Divine Disease, which was the most terrible of the ailments associated with the Blight. The infection was created by Dagoth Ur and is spread by contact with a corprus beast or by specific curse. It destroys the mind of the sufferer and grotesquely deforms the body. It is not known how the transformation of Sixth House cultists into Ascended Sleepers and other monsters is related to Corprus. Some of the physical effects are similar, but the cultists remain lucid and attain a higher spiritual state whereas more mundane sufferers become mindless husks, like the common undead raised by necromancers.
Perhaps the most frightening aspect of this form of corprus is that it is completely incurable. Those infected were sent to Tel Fyr to live in the Corprusarium, where they were treated and studied by Divayth Fyr, a 4000-year old wizard working to uncover the nature of the disease and create a cure. Fyr had a theory that the Divine Disease is in fact not a disease, but rather a divine blessing that most mortals can't handle, pointing to the fact that the victims are completely immune to other diseases and don't age.
The last known Dwemer, Yagrum Bagarn, was infected with corprus atop Red Mountain during his exploration of Tamriel. He sought refuge at the Corprusarium in Tel Fyr, where he offered his knowledge and provided theories on the disappearance of the Dwarves.
In 3E 427 the Nerevarine became infected with corprus, and Divayth Fyr provided a "cure" for the disease. This "cure" did not actually cure the disease but rather removed the negative effects, such as crippling physical deformities and insanity, while retaining the positive effects, including immunity to all diseases, increased strength and endurance, and possibly also immortality.
Yagrum Bagarn, the last Dwemer, who was infected by Corprus
For game-specific information, see the Morrowind article.