Essentially, after the fall (or falls) of Napoleon, the allies (Britain, Austria, Prussia, Russia, Sweden and various other European states) reinstated the French monarchy, believing it to be more conducive to European peace than continued Napoleonic rule or the reinstatement of a Republic.

In more detail:

The 'French Revolution' is a descriptor we use for ten years of French history from 1789 to 1799, which covers a period of reforming monarchy (broadly, 1789-1792), a period of Jacobin rule and 'Terror' in the Republic (1792-1794), and Republican rule under the Directory (1795-1799). The coup in 1799 wherein several conspirators (including Napoleon Bonaparte) gained control of the French state and began to institute the Consulate (which Napoleon would end up leading) is generally taken to be the end of the Revolution.

The French Monarchy was abolished in 1792, and Louis XVI was executed early in 1793. After twelve years of the Republic in its various forms, France was made into a hereditary empire under Napoleon in 1804. The Empire lasted for ten years, until Napoleon was forced to abdicate by his Marshals as part of a peace deal with the allied armies that has pushed Napoleon back through Europe and invaded France. (Napoleon would return in 1815 and attempt to re-establish his rule of France, but his defeat at Waterloo ended that possibility) The key part of this potted history to this answer is that, from 1792 to 1815, France was almost constantly at war with other European powers, and in 1792-95 and 1813, 14 and 15, it was at war with pretty much all of the major European powers and many minor ones.

At this point, the Allies, in accordance with the formal alliance they had made in 1813, sent representatives (or their monarchs went) to Vienna, to discuss and build a postwar order in Europe that would avoid further bloodshed, establish a balance of power between the great European powers, and restrain the incipient forces of nationalism and democratism that had, in the eyes of several of the Allies (principally Russia and Austria) caused the twenty years of near-constant war that Europe had been afflicted with. This is known as the Congress of Vienna.

One of the easiest steps the Congress could take was to restore the Bourbon monarchy. While in 1814 France was exhausted and war-weary, it could still pose a threat to peace (as Napoleon's return in 1815 proved), and the Allies felt that both a Republic and a Bonapartist Empire would lead to war - after all, they had in the past. A restored Bourbon monarchy, on the other hand, would likely not go to war - for one thing, the Bourbons had understood the concept of European status quo, and were diametrically opposed to the Jacobin and Nationalist ideas that the Congress was trying to restrain. Thus they reinstituted the Bourbon monarchy.

It should be noted that the restored monarchy did not work in the same way as Louis XVI's ancient régime had - the France Louis XVIII ruled was the same state Napoleon had built - and indeed that state is still the foundation of modern France. In many ways, Napoleon and the restored Bourbons were far more powerful monarchs than Louis XVI ever had been, able to censor the press, levy taxes and conscription, and understand and control their state far more comprehensively than the ancien régime had been able to. Several of these issues, particularly press censorship, would play into the 1830 Revolution.

The Revolutions in France were also not over - Charles X was replaced as King by Louis-Philippe d'Orleáns, who would himself be deposed by a revolution in 1848. The short-lived Second Republic would then elect Louis-Napoleon Bonaparte (Napoleon I's nephew) as President, and he would institute a second hereditary Empire as Napoleon III, which would only be brought down by Prussia in 1870. The Third Republic that emerged from that war was the longest-lasting French republic, and the beginning of the unbroken period of Republican government in France.