The poem describes the importance of cats having three different types of names - an everyday name used by the family, fancier names if they sound sweeter, and a name that is particular and more dignified to help the cat keep their tail up and pride. It provides examples for each type of name, such as everyday names like Peter or George, fancier names like Plato or Electra, and more dignified names like Munkustrap, Quaxo or Bombalurina.