The aristocracy is made up of feudal monarchs—you know, all those the medieval kings and queens we love to watch on TV and read fantasy novels about. Under feudal rule, society was made up of, in Marx's words, "feudal lords, vassals, guild-masters, journeymen, apprentices, serfs" (Section1.3).
What's important to understand about the aristocracy, according to Marx, is that their economic system, feudalism, was outmoded by the new technology and market expansion that grew with capitalism. "The feudal system of industry [...] now no longer sufficed for the growing wants of the new markets" (Section1.8).
The aristocracy was doomed to the dustbin of history because economics and class struggle, for Marx, drives progress. "The world-market [...] pushed into the background every class handed down from the Middle Ages" (Section1.10).
In other words, controlling a bunch of land and having serfs farm for you was no longer enough to keep the increasingly successful bourgeois merchants under the aristocracy's thumb. For one thing, the bourgeoisie wasn't going to sit around believing in the divine right of kings while their successful growth of capital handed them more and more political power.
Let Them Eat Cake
That didn't stop the aristocracy from trying to maintain their place on top. They sometimes collaborated with the bourgeoisie to keep their own nobles in check (Section1.12)—but sometimes they appealed to the proletariat for help against the bourgeoisie (Section3.1-9), too. According to Marx, "The aristocracy, in order to rally the people to them, waved the proletarian alms-bag in front for a banner" (Section3.4).
But the offerings didn't fool anyone, because "the people, so often as it joined [the aristocracy], saw on their hindquarters the old feudal coats of arms, and deserted with loud and irreverent laughter" (Section3.4).
The aristocracy's fatal error, Marx says, was that it didn't understand how progress works: economics and class struggle drive history, and by the time the bourgeoisie had control of modern industry, it was too late to turn back the clock. Even though the aristocracy wrote "bitter, witty and incisive criticism [of the bourgeoisie], striking the bourgeoisie to the very heart's core," their efforts were "always ludicrous in its effect, through total incapacity to comprehend the march of modern history" (Section3.3).
The Aristocracy's Timeline