The description "flat character" is used to describe those characters in a story that are not given any depth of development. The reader generally meets such characters only briefly, and sees only one aspect of each flat character's personality.
In "Rikki-Tikki-Tavi," Karait is a flat character. There is no information supplied about Karait beyond what is essential for the purposes of the plot. The reader knows that Karait is a poisonous "dusty brown snakeling" and that he becomes Rikki's first conquest.
Darzee's wife is a flat character. The reader learns that she is "a sensible bird" and reads how she helps Rikki trick Nagina, but nothing else. Darzee's character is developed in more detail, as the reader learns that he is something of a coward, that he "could never hold more than one idea at a time in his head," and that he loves to make up ballads celebrating the feats of others - so Darzee is not a flat character.
Chuchundra the muskrat and the Coppersmith bird could be considered other flat characters. They each serve one specific role in the story.