Short answers, pulled from the story.
The House of Bourbon originated in 1272, when Robert, Count of Clermont, the sixth and youngest son of King Louis IX of France, married Beatrix of Bourbon, heiress to the lordship of Bourbon. Their son Louis was made the first Duke of Bourbon in 1327.
The House of Bourbon became rulers of France when Henry III of France was assassinated on the 31st of July 1589, ending the Valois line. Henry of Navarre, as the senior surviving representative of the Capetian dynasty, claimed the French throne as Henry IV. He was crowned at the Cathedral of Chartres on the 27th of February 1594.
Bourbon monarchs ruled Spain from 1700, Naples from 1734, Sicily from 1735, the Kingdom of the Two Sicilies from 1816, Parma from 1731, and Navarre from the 16th century. Today the Bourbon line continues on the thrones of Spain and Luxembourg.
The Bourbon monarchy in France was abolished on the 21st of September 1792 during the French Revolution, and Louis XVI was executed on the 21st of January 1793. Restored in 1814 and 1815, the senior Bourbon line was finally overthrown in the July Revolution of 1830. The Orleanist cadet branch then ruled until it too was overthrown on the 24th of February 1848.
Henri, Count of Chambord, refused the restored throne because he insisted France abandon the tricolour and restore what he regarded as the true Bourbon flag bearing the fleur-de-lis. The French National Assembly could not agree to that condition, and a provisional Third Republic was established instead. Henri died without children in 1883, ending the main French Bourbon line.
The House of Bourbon continues to reign in Spain under Felipe VI and in Luxembourg under Guillaume V. The dynasty is considered the oldest royal dynasty in Europe still existing in the direct male line, with Capetian male ancestors traceable to Robert of Hesbaye, who died in 807.