Flat characters—minor figures that play a supporting role in the story—are used to move the plot along, providing subtle exposition. Such characters do not undergo substantial change or growth in the course of a story. They are also known as "two-dimensional characters" or "static characters."
While main characters need to be believable to the reader, with a full range of personality traits and personal issues revealed, flat characters have a much narrower purpose and aren't developed to the same extent as a story's protagonist.
Examples of Flat Characters
Good examples of flat characters are Crabbe and Goyle in the Harry Potter series. Telling their backstory, exploring their motivations, and having them change over time would make it almost impossible to follow the story arc. Rather than attempting to flesh out these characters, author J.K. Rowling makes them "two-dimensional," or flat. Crabbe and Goyle are unintelligent, sycophantic followers—necessary to the plot, but otherwise unimportant.
A more classic example is the character of Mr. Collins in Jane Austen's "Pride and Prejudice." He serves a vital role in the story for a character who is almost stereotypical in his pomposity, egocentricity, and class consciousness. Mr. Collins is at the center of a plot point through which protagonists Elizabeth and Darcy get together, providing a necessary comic element. But his character stays essentially unchanged; in fact, his lack of responsiveness is part of what makes him funny.
Creating Flat Characters
Creating flat characters is relatively easy; the writer needs to develop them only to the extent that the reader understands that character's relationship to the main characters. Whatever interaction that minor character has with the main character needs to be believable, either moving the protagonist toward a goal or set him back.
These figures in your writing are usually born of a single emotion. A flat character who is fearful might encourage the protagonist to be brave. One who is hedonistic might make the main character indulge where she shouldn't. These people serve a purpose in your story, and when the main characters have many different emotions and arcs, these characters are always stable.
Uses in Different Types of Fiction
Flat characters are a staple of genres that require particular personality types, like romance or young adult fiction. Typically, the less literary the novel, the less you need to disguise such characters. Some examples:
Few readers want to see reliably evil characters grow a conscience or feel guilt as a result of their actions. Flat characters are used extensively as henchmen who, unless they are one of the key players, almost never change. This makes it easier to understand and gives the reader a more comfortable sense of catharsis when they are subsequently dispatched, as often happens in graphic novels, adventure tales, and westerns.
Readers of series like "Twilight" choose the next book in the series because they want to revisit the characters they know and love. Authors of such series must be careful to avoid changing their characters too radically and undermining their popularity.
Stock characters, such as the wise adviser in the "Quest" fantasies, are intended only to further the plot by providing critically important information to the protagonist. Flat characters in the fantasy genre are appropriately called "quest-givers." Should such a character change, that person's role in the story would come into question.