As the first answer suggests, most of Twain's satire regarding nobility relates to the Duke and King. Here are a few specific examples.
1. The King and the Duke are con artists by profession, illustrating Twain's belief that "nobles" are simply normal people who are able to convince others to respect them (usually through false pretenses) and to offer them money or other goods.
2. The King and the Duke prey upon the ignorant just as nobility often tries to keep the lower class illiterate and uninformed. While the Wilkes girls are certainly not illiterate, they and their fellow townspeople are naive and ignorant when it comes to other cultures and customs. Because of their ignorance, the "nobility" is able to once again take advantage of them.
3. The Grangerfords and Shepherdsons represent the closest thing to royalty or nobility that the American South possessed during the pre-Civil War time period; and yet, they are some of the most uncivilized characters in the book. Twain uses them to show that "breeding" and social class do not equal morality. Both families act barbarically, resulting in an almost complete annihilation of a family.
4. The Grangerfords also serve as a satirical attack upon nobility in that they own slaves, forcing Jim to hide from them as Huck enjoys his time with Buck Grangerford. Instead of representing what is noble and good in society, the Grangerfords continue to participate in the barbaric practice of slavery.
5. In the end, when the Duke and the King get their comeuppance, it is almost entirely because of a small child (Huck). Twain seems to be implying that nobility is not as infallible or untouchable as it thinks, and that their abuse of power does not have to continue.