A flat character is one that has only the bare minimum of characteristics necessary to play their role in the story.
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 her medical bills, all while trying to polish off her doctoral dissertation on Ming-era Chinese poetry, squeezing 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 relayed, the customer waiting for their 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. Nominal Importance is another example of this.
Characters who start out flat can be fleshed out into Rounded Characters with Character Development, Hidden Depths and/or being Rescued from the Scrappy Heap. Or some characters can look flat until the Red Herring Shirt reveal shows that they are truly full characters. 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. Some writers intentionally make characters flat to display their unhealthy psyche.
For more fleshed out examples (for lack of a better term), see The Generic Guy. If you were looking for tropes about characters that are literally flat, see Paper People, Squashed Flat, or maybe Petite Pride or A-Cup Angst.
For emotionally flat characters, see The Stoic.
As this trope can be very subjective among people, please do not add any examples.