The Leopard is the official symbol of the House of Salina, the aristocratic family whose decline is really the main focus of this book. On several occasions, the book even compares its main character (Prince Fabrizio) to a leopard. He's strong, but also graceful… like a leopard.
The novel closes with Fabrizio's daughter, Concetta, thinking the she's seen a leopard running in the corner of her eye. But it's only an old stuffed dog getting thrown into the garbage, which is kind of what history is doing with the House of Salina. Womp womp.
One of the sad things about the leopard in this book is that its gracefulness and power are being replaced by the "new men" of Italy, who the book compares to jackals and hyenas. These people aren't graceful at all; they're scavengers and opportunists looking to exploit any situation to make a quick buck. Don't get us wrong— the fall of the aristocracy is basically a good thing— but Lampedusa shows us the seedy side of the hustling required to get ahead in politics. Democracy, after all, gives us Frank Underwood.
Oh, and fun fact about The Leopard; it should be called The Ocelot or The Cheetah. That's what the Italian Il Gattopardo translates to. But since an ocelot or cheetah just doesn't connote the same beauty and grace as a leopard (we might think of Chester Cheeto instead of this graceful dude), it's better symbolism.