Mentioned by Desmond when he questioned Jack and Locke.
Aaron contracts an illness that, though Claire is concerned that it might be the sickness Rousseau's crew contracted, clears up in a couple of days.
When she was abducted by Ethan, Claire was given injections of CR 481516-23 42 into her womb. This may have a connection to the sickness, since the substance CR 481516-23 42 is claimed to be a vaccine.
Charlie's actions in the episode Fire + Water could be the first signs of The Sickness.
The sickness is what has made the the Others the way they are.
Possibly caused by an electromagnetic force
The sickness does not actually exist. This is being used as social experiment. The Others used it to convince Claire (unsuccessfully) and Rousseau to give up their babies. They also used it to convince Desmond, etc to stay inside the hatch. Rousseau was further convinced to kill her team, convinced they had The Sickness.
Who believes in "The Sickness"
Rousseau strongly believes in it (in an almost paranoic fashion).