Bourbon was a province in France. In 1287, Bourbon passed to a woman, Beatrice, who married King Louis IX's son Robert (one of his younger sons who would never inherit the throne). Their son, Louis, was made the Duke of Bourbon (previously a lordship). They spent a couple of centuries as influential but not terribly impressive Dukes in France.

However, that French line of succession came in when Henry III was assassinated and the Duke of Bourbon became Henry IV (he was already King of Navarre). Henry IV was fascinating. He was a Huguenot who was a target in the St. Bartholomew's day massacre (he converted back to Catholicism to avoid it and renounced it later). But none of that is relevant to your question. The important thing is that with Henry IV the Bourbons became kings of France. Henry IV was eventually assassinated but his line lived on. The Bourbons were the kings on the throne during the French Revolution. And the kings brought back when the monarch was restored (and then thrown off again this is something of a pattern with them).

When poor disabled Charles II, the infamously too inbred Haspburg, of Spain died he left the Spanish empire to his grand-nephew. His grand-nephew happened to be the grandson of King Louis XIV of France, a Bourbon. This set-off the War of the Spanish Succession as no one wanted the Bourbons to have that much power.

During Charles II's lifetime his territories included Sicily, Naples, and Sardinia. The conclusion of the War of the Spanish Succession led to the Bourbon, Phillip V, on the throne of Spain. But most of the other European territories had to be given away including Naples and Sicily (and he renounced any claim to the throne of France thus the second House of Bourbon was born).

Phillip V married the niece of the Duke of Parma (who was his second wife), who definitely had Italian ambitions. And the two of them set their sights on winning back that lost territory in Italy. They occupied Naples and Sicily and also had their son, Charles, made Duke of Parma. After a war with Austria (which was sparked in Poland) Charles was officially king of Sicily and Naples. Although Parma was ceded to the Austrians.

So Charles was the first Bourbon king of Sicily and Naples, where he lived (rather novel for kings of Sicily and Naples). He was widely regarded as a good king. Famously, he was king when Pompeii was discovered and was very excited about the excavation and all of the new discoveries. He also did great things for the economy, much needed in an area that had been fought over and neglected for so long.

After his elder brother died Charles was made king of Spain, Charles III. Eventually, he ceded control of Naples and Sicily to his son, Ferdinand. Thus creating the third House of Bourbon ruling in Europe.

The Bourbons, this third line, continued to rule the Kingdom of the Two Sicilies, which was the largest unified territory in Italy, until Napoleon forced them out. And then took control back after the fall of Napoleon until the unification of Italy in 1860, which is where your book comes in.

Incidentally the current king of Spain, Felipe VI, is a Bourbon. They were a tenacious group. They gained power partially through luck (unlikely inheritances of major thrones falling into their lap twice). Through military power. And through savvy political alliances. Until the French Revolution, they remained closely aligned with the French Bourbons. They were the great rival to the Hapsburgs in their day.