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"She runs the gamut of emotions from A to B."
A flat character is one that only has the bare minimum number of characteristics necessary to make them interesting enough to carry off their purpose. Flat characters with just one defining trait are known as one-dimensional characters, while flat characters with a few more traits are two-dimensional characters.
One example of a flat character is the pawnbroker in A Christmas Carol, who buys Ebineezer Scrooge's bed curtains from the maid who stole them while he was lying on his bed, freshly dead. The scene with the pawnbroker consists of characters that are all so flat - other thieves selling things, the maid, the pawnbroker himself - that some productions of the book have dropped the entire scene, with little loss in the telling of the story.
It might be simpler to just have a look at Characters As Device, to cut through all the fog of lit-crit jargon around characterization.
Being flat is not automatically bad. Character depth should be proportional to the character's importance to the story. The fact that the cocktail waitress is a leukemia survivor who is working two shifts to pay off medical bills, all while trying to polish off her doctoral dissertation on Ming-era Mandarin poetry and squeeze out enough time to decide which of her three suitors will best be able to get along with her aging, beloved Pomeranian-Pug pup all comes under the heading of "too much information." By the time all that is related, the customer waiting for his drink has died of thirst.
Indeed, adding details to the character indicates to the audience that the character is to be important. The Spear Carrier, the Red Shirt, the Bit Character may require a Flat Character, to prevent the reader from feeling cheated. This is why we get the Fatal Family Photo - if an otherwise interchangeable Red Shirt takes the time to establish his hopes and dreams, it's obvious they're going to be dashed in the name of drama.
Characters who start out flat can be fleshed out into Rounded Characters with Character Development, Hidden Depths and/or a Rescue From The Scrappy Heap. They can also become a Static Character trapped in amber with repeat uses of a Reset Button or Snap Back, negating what little growth they manage; and they may mutate into another sort of Flat Character with Flanderization.
For more fleshed out examples (for lack of a better term), see The Generic Guy. If you were looking for the trope about characters that are literally flat, see Paper People, or maybe Petite Pride or A-Cup Angst.