Napoleon III was deposed as emperor in 1870 following the his personal defeat by the Prussians at Sedan, and the general defeat of France. The Bourbons were offered the throne, but refused what would have been a constitutional monarchy. So but for the hard headedness of the would be Henry V, France would have been a monarchy at the onset of the First World War.
The current heir to the French throne, quoted here, is from the House of Orleans. They last held the throne until deposed by the ‘Occupy Paris’ crowd in 1848.
In 1871 the French elected a great majority of royalists to the Assembly, but they were split between Legitimists and Orleanists. Unable to agree on a King, they sort of drifted into a republic by default.