A play on words is a literary technique where words are used in an unusual way, putting emphasis on the words themselves in an unconventional manner. A pun, which is when an alternate meaning of a word is exploited with a humorous intent, is an example of a play on words.
In the example about which you are asking, Nadia uses two words which are almost identical to create an interesting literary effect. In response to Noah's facetious comment that her dog, Ginger, "is the dog that invented E = mc2", Nadia replies haughtily,
"Ginger is a genius of her genus. She is the best there is of Canis familaris, and Alice is the best of her litter".
The play on words here involves two words which differ in spelling by only one letter, genius, and genus, which means "species". Nadia, who always asserts that "Ginger is a genius", is qualifying her statement, emphasizing that Ginger is indeed very smart, for a dog. As Noah and everyone else should well know, Nadia is not so arrogant as to claim that Ginger is smarter than a human genius such as Einstein, the discoverer of the formula E = mc2. Nadia is only saying that, among the world of dogs, Ginger's genus, or species, Ginger is far smarter than average, or, in other words, a genius, and that Alice, the puppy that Nadia wants to give to Julian, is the best of her litter (Chapter 3).