An exemplification is an illustration or classic example of something. Breaking into loud, operatic singing at the breakfast table might be the perfect exemplification of your family's strange sense of humor.
The grace and beauty of a ballerina dancing on a stage is an exemplification of the principles of ballet itself — in other words, she is a living example of what ballet should be. When you illustrate something this way, by showing a concrete example, you exemplify. And the act of using that example is exemplification. The Latin source of both words is exemplum, "example or model."