Ostensibly used to refer to a young and impertinent person by someone who is presumably older and wiser, "whippersnapper" is a word that is rarely used non-ironically in everyday life. In books, movies, and other fictional works, the word is used more to denote the stuffiness and old-fashionedness of the speaker than the qualities of the person he or she is describing as a "whippersnapper".
In the 17th century, the phrase "whip-snapper" referred to inexperienced young cowboys who stood on the sides of streets, passing the time by (among other things) snapping whips, possibly to call attention to themselves. Snapping a whip is the most basic skill learned by a novice cowboy, and so those who were unable to perform more complex tasks nevertheless showed off their prowess by displaying their impressive whip-snapping skills. A phrase that already existed to describe idle and lazy young men, "snipper-snapper", was conflated with "whip-snapper" to denote a beginner who, though unskilled, thinks much of his abilities. Eventually the word evolved to mean someone insignificant who is presumptuous and arrogant enough to annoy those old enough to know better.