The monarchy was overthrown in the autumn the year before, materially in August and officially in September. Before that, war had broken out between France and multiple different states. The purpose, as far as France's enemies were concerned, was to restore the monarchy to its former power. The wars that followed had more or less the same end-goal in mind until they succeeded in overthrowing Napoleon Bonaparte, who was the successor to the Revolutionary government. At that point, 1814, France had already been a hereditary monarchy since 1805, with Napoleon as emperor. The Bourbons took power in France for the first time since 1792. They managed to hold onto it for a little less than a year. Napoleon made a brief comeback in 1815, before being defeated at Waterloo and forced into exile for good.

(Back to 1789 a bit)

When the Revolution broke out, many nobles and royals left the country. Among them were Louis XVI's two brothers. The émigrés joined forces with the foreign enemies of the Revolution. After Louis XVI was executed in 1793, the royalists viewed his only son as the new king, despite the abolition of the monarchy, and their goal was to put him on a restored throne. Louis XVII died two years later, which left his uncle as king Louis XVIII.

When Napoleon was forced to abdicate, Louis XVIII took the throne and reigned as a constitutional monarch for ten years. Lived happily ever after, more or less. (Not so much for his brother Charles X who was overthrown in another revolution in 1830. He was replaced by a relative called Louis Philippe, who was in turn overthrown in 1848. Then there was another republic, followed by another Bonaparte and another empire. Until Napoleon III (Napoleon II was never ruler, like in the case of Louis XVII) was overthrown by another invasion, though the heir to the old monarchy didn't get the crown due to political differences. Instead there's the Third Republic, and no more monarchs since then.)

After the "defeat" of the Prussian army during the Battle of Valmy, the Republic was proclaimed and the monarchy of France was cast aside. Naturally this would cause people to assume that the French monarchy was no more but as a result of the Terror and general nobility flight from France during the early years of the Revolution, many nobles ran to Prussia and Austria including Louis XVIII.

As a result of the defeat of Napoleon, the governments of Europe were interested in pretending that the Revolution never happened and aimed to try to continue the Bourbon reign in France for the sake of continuity as well as he had the legitimate claim on the throne of France.

Yes, it was overthrown in 1789, but Louis XVI's brother, Louis XVIII, spent years in exile waiting to be put back on the throne. (Louis XVII, by the way, was Louis XVI's son and died very young while being held prisoner.)

After Napoleon was finally defeated in 1814 by the Sixth Coalition against him, he was forced to abdicate his throne and sentenced to exile as the ruler of the small island of Elba.

The Coalition powers were conservative and wanted to maintain the status quo. They also feared liberalism—the movement demanding suffrage for more people, freedom of trade, and the abolition of the privileges of the nobility—as a destabilizing force. As a solution to both of these problems, they decided to put Louis XVIII on the throne of France. This was called the Bourbon Restoration, since it brought the Bourbons back into power.

However, the British especially were not in favor of making him an absolute dictator, so he was pressured to issue the Charter of 1814, which guaranteed basic rights to the people. More symbolically, the Bourbon Restoration abolished the tricolour and brought back the traditional white flag (no snickering), sometimes with gold fleurs-de-lis to make it more distinctive.

After only a year on Elba, Napoleon broke out and quickly gained enough popular support to rule France again for the famous Hundred Days (really 111). But then he was defeated at the Battle of Waterloo, abdicated again, and Louis XVIII was put back on the throne, which he held until his death.

Napoleon was exiled again, this time not to a nice place like Elba, but to St. Helena, a tiny island truly in the middle of nowhere: 1200 miles from the nearest landmass. The second time around, the Coalition really wanted to make sure that he would not come back.

I won't get into the details (someone else can), but the Bourbon dynasty would finally fall in 1830, to be replaced by its more liberal cadet branch, the house of Orleans, in the July Revolution. (Which, by the way, is the revolution depicted in this painting.) And then in 1848, the Orleans fell, too, when France became a republic again.