A plague ravages the kingdom, setting the Adventurers on a quest to find a cure. An adventurer emerges from an ancient tomb, unopened for centuries, and soon finds herself suffering from a wasting illness. A Warlock offends some dark power and contracts a strange affliction that spreads whenever he casts Spells..
A simple outbreak might amount to little more than a small drain on party resources, curable by a casting of Lesser Restoration. A more complicated outbreak can form the basis of one or more Adventures as Characters Search for a cure, stop the spread of the disease, and deal with the consequences.
A disease that does more than infect a few party Members is primarily a plot device. The rules help describe the Effects of the disease and how it can be cured, but the specifics of how a disease works aren’t bound by a Common set of rules. Diseases can affect any creature, and a given illness might or might not pass from one race or kind of creature to another. A plague might affect only Constructs or Undead, or sweep through a Halfling neighborhood but leave Other Races untouched. What matters is the story you want to tell.