Short answers, pulled from the story.
A delusion is a fixed belief that is not amenable to change in light of conflicting evidence. It differs from a belief based on false or incomplete information, because people holding those other beliefs can readjust once they review the evidence.
Delusions are categorized into four groups: bizarre, non-bizarre, mood-congruent, and mood-neutral. French psychiatry also distinguishes the paranoid delusion, seen in schizophrenia, from the highly systematized paranoiac delusion, seen in paraphrenia.
Persecutory delusions are the most common type. They involve being followed, harassed, cheated, poisoned, conspired against, or spied on, and require the belief that harm is occurring and that the persecutors intend to cause it.
Karl Jaspers, a psychiatrist and philosopher, defined the four main criteria for a delusional belief in his 1913 book General Psychopathology. The criteria were certainty, incorrigibility, impossibility or falsity of content, and not being amenable to understanding, of which only the first three remain in the DSM-5.
The Martha Mitchell effect describes a true belief mistakenly classified as delusional. It is named for the wife of the attorney general who alleged illegal activity in the White House, claims thought to signal mental illness until the Watergate scandal proved her right.
Delusions are often treated with antipsychotic medication, which reduces symptoms with a medium effect size according to a meta-analysis. Cognitive behavioral therapy and metacognitive training also reduce delusions relative to control conditions.