Between the pride and intellectuality of his mother and the sensuality and irresponsibility of his father, poor Prince Fabrizio lived in perpetual discontent under his Jovelike frown, watching the ruin of his own class and his own inheritance without ever making, still less wanting to make, any move toward saving it. (1.12)
Here's the central conflict of this story: Fabrizio sees his entire social class going into the toilet of history, but he can't bring himself to do anything to stop it. His two main reasons for doing nothing are: 1) He can't stop history even if he wanted to, and 2) There's not much about the lazy, entitled aristocracy that's worth saving. After all, it's hard to argue that you deserve all the money and power in the world because you were born into the right family.
"For the King, who stands for order, continuity, decency, honor, right; for the King, who is sole defender of the Church, sole bulwark against the dispersal of property." (1.22)
When asked what a young Sicilian soldier has died for, Fabrizio's friend answers that the young man died for order and decency, which are just general concepts that don't really mean a whole lot to Fabrizio. This friend is obviously just anti-democratic, meaning that he doesn't think the opinion of the masses should decide important issues. That, plus he doesn't want all the aristocrats' property to be divided up more equally among people. That's what you call looking out for yourself.
The Prince gave a start of annoyance; so touchy is the pride of class, even in a moment of decline, that these orgiastic praises of the beauties of his future niece offended him. (3.52)
Fabrizio knows his class is dying, but that doesn't mean his pride is completely gone. He still gets really annoyed when people speak a little too candidly to him. It's as if he's saying, "Hey, I'm still entitled to a little respect!"
"How foul, Excellency! A nephew of yours ought not to marry the daughter of those who're your enemies who have stabbed you in the back! […] It's the end of the Falconeris, and of the Salinas too." (3.54)
Fabrizio's friend Ciccio is aghast when he learns that Fabrizio's nephew Tancredi is going to marry the daughter of the mayor of Donnafugata. For Ciccio, the aristocracy should never mix with the commoners, no matter how rich these folks might be. On top of all that, he blames democratically elected idiots for pushing the elegant, intelligent aristocrats out of power.
"[Till] I came along we'd been an unlucky lot, buried in the provinces and undistinguished, but I have the documents in order, and one day it will be known that your nephew has married the Baronessina Sedàra del Biscotto." (3.82)
Don Calogero is sensitive to the fact that his daughter is marrying a man from a noble family (Tancredi). To show his respect, he tells Fabrizio that his daughter is actually descended from a noble family, too, and that he can secure the necessary papers to prove it. Turns out that this guy has a little more social tact than anyone's willing to give him credit for.
[Free] as he was from the shackles imposed on many other men by honesty, decency, and plain good manners, he moved through the jungle of life with the confidence of an elephant which advances in a straight line. (4.1)
In a way, Fabrizio respects Don Calogero for not caring about all the mannerisms that aristocrats care about. Don Calogero marches through life clumsily and ignorantly, but he always gets where he's going, kind of like an elephant marching through the jungle.
[But] from that moment there began, for him and his family, that process of continual refining which in the course of three generations transforms innocent peasants into defenseless gentry. (4.5)
As Lampedusa tells us, Don Calogero doesn't realize what he's getting into when his daughter marries Tancredi. Over time, her life of luxury and idleness will make her family just like the Salinas. They'll all get totally lazy and complacent, and they'll end up losing everything to more competitive people. It's a good point to make, but Lampedusa is also kind of showing off the fact that he knows everything that'll happen for a hundred years after this story.
"I am a member of the old ruling class, inevitably compromised with the Bourbon regime, and tied to it by chains of decency if not of affection." (4.93)
When offered a position as a senator in the new Italy, Fabrizio refuses because he's just not competitive enough to muck around with all of Italy's social climbers. He'd just as soon get out of the game altogether and fade into the background of history.
"They're just different; perhaps they appear so strange to us because they have reached a stage toward which all those who are not saints are moving, that of indifference to earthly goods through surfeit." (5.19)
For Father Pirrone, aristocrats are like a completely different type of human. They have no interest in money or property, but only because they've never lived without these things. That's like standing on a boat while someone drowns and saying to him, "Meh, oxygen is overrated."
"[If], as has often happened before, this class were to vanish, an equivalent one would be formed straightaway with the same qualities and the same defects; it might not be based on blood any more, but possibly on… on, say, the length of time lived in a place, or on greater knowledge of some text considered sacred." (5.29)
Father Pirrone is pretty cynical when it comes to social classes. He knows that the moment Fabrizio's class vanishes, another one will rise to take its place. This new class will just have a different standard of measurement for "greatness," like how long a person has lived in a town or how much they know about a certain book. Or in other words, he believes that the world will always have snobs and cliques, which are just part of human experience.