Elaboration
By my understanding, a genus is a wider class of things to which a particular thing belongs. A differentia is the differentiating attribute(s) which distinguish(es) that type of thing from the wider class to which it belongs.
For example, a frog is an amphibious animal. The differentia in this case is "amphibious," and the genus is "animal."
Although this preliminary definition incorporates a differentia and a genus, and might function satisfactorily for a young child, a person might learn that there are many other amphibious animals other than frogs. Such a person might satisfy needs for deeper clarity and articulation by then formulating a more specific, genus-differentia definition.
A much more specific definition of "frog" could be: "Any of numerous tailless, aquatic, semiaquatic, or terrestrial amphibians of the order Anura and especially of the family Ranidae, characteristically having a smooth moist skin, webbed feet, and long hind legs adapted for leaping."2
In this more specific definition, broadly speaking, the genus is "amphibian," and the differentia includes all the attributes that together distinguish frogs from other amphibians.
Notes
1Although this definition and the elaboration I've provided are mine, I'm grateful for inspiration that I drew while formulating it from both Ayn Rand's nonfiction book, Introduction to Objectivist Epistemology, and from David Kelley's book, The Art of Reasoning.
2"frog," The American Heritage Dictionary of the English Language, 4th ed. Boston: Houghton Mifflin, 2000. www.bartleby.com/61/. [June 30, 2005.]
