An illness is what you have when you are sick. This is always a bad thing and ranges on a scale of unpleasantness from the bother of a runny nose to the horror of bits falling off. Most illnesses are caused by an infectious agent (not to be confused with PPC agents) or an inherited condition. Many illnesses can and do result in death, some quickly, some slowly.
Many agents view Mary Sues as an infection and themselves as the cure.
In Fanfiction Edit
Illness is a reasonably common plot device in fanfiction. It's something we all experience at one time or another, so we can all relate to it. Fanficcers may write about characters getting sick for any of several reasons, including drama, pathos, horror, hurt/comfort (which may or may not lead to romance), and humor.
Afflictions used in fanfiction can range from the familiar (cold or flu) to the outlandish (fantasy pox, space sickness). For some reason, pneumonia is common in the the X-Files fandom. A well-written illness, even a made-up one, should have defined symptoms, a clear progression, and courses of treatment that make logical and medical sense—unless it's being played for laughs, in which case the Rule of Funny applies.
A fic centered on illness is referred to colloquially as a "sick fic."
In Badfic Edit
In badfic, illness tends to take the form of either a real illness portrayed incorrectly or a made-up illness that can't be defined, a la the Incurable Cough of Death, Soap Opera Disease, or Victorian Disease. They may also be guilty of bad biology by having a character get a disease they shouldn't (e.g. a Tolkien elf catching a pathogen).
If the disease is vague or nonsensical, the cure is likely to be equally vague or nonsensical. If there is no cure, rest assured that the sick character will nevertheless be healed by a magical deus ex machina that turns up at the last moment. Either way, you've got a case of bad medicine on your hands.
Sometimes, sick fics are written as an excuse for shipping, so that they can have one half of their ship take care of the other half. The more contrived the scenario that forces the two characters together, the bigger the charge. Bad sick fics are often guilty of making either the sick character, the carer, or both act out of character. The sick character may act like a whiny baby or neglect their health in ways they wouldn't if they were in character. The caring character may act like a fussy mother hen and smother the sick character, even if they're normally reserved or downright evil. These stories sometimes have the characters kiss, or even have sex. This is a Very Bad Idea if it would risk the sick character's recovery or if it would result in the caring character being more exposed to germs than they already were by being in close proximity to the sick character. Bonus points if the sex heals the sick character.
The idea of having sex while sick, up to and including getting off on kinks involving symptoms of illness, is squicky to a lot of people. Kinkfic of this nature, while not necessarily badfic, would have to be extremely well-written to convince the average reader that their favorite character revels in something that grosses them right the heck out. The more uncommon the kink, the harder the sell.