Who was Napoleon III? Did the great powers re-install the French Monarchy after Waterloo? What exactly happened to France after the surrender of Napoleon I from the years 1812-1850?

For a more in depth look, I would recommend Paris Between Empires, 1814-1852. It's basically a six hundred page book of what I'm going to say with more details about how Napoleon III came to power (as it is a weak point).

Let's start in the dark days of 1814. Napoleon is still in power, leading France in a defense against the Allies invading from all points; from the South West comes the perifidious albion, Wellington leading the Anglo-Portuguese army and pushing aside the venerable Soult; from the South East, the Kingdom of Italy is holding off Austrians and Neapolitans pushing against the French puppet kingdom as the brave Eugene de Beauharnais holds them off with a skill similar to his father-in-law; to the East and North East, the main Allied armies push into France with Napoleon leading the most valiant defenses he could muster, defeating army after army but unable to kill the hydra. However, the French people are tired; recruits aren't reporting for their conscription and the generals want to live off the fat they've earned from nearly twenty years of constant war so Marshal Ney represents the Marshalate and tells Napoleon that they are done as is the army but Napoleon refuses saying that the Army will follow him, Ney responds that they will follow their generals.

Napoleon abdicates in favor of his son in mid-April but the Allies refuse it as they had taken Paris. Napoleon, sad and ruined, accepts the new terms of absolute surrender. France is happy for once, the wars of Napoleon and the Revolution were over and hopefully commerce would return. The Allies return a Bourbon to the French throne, Louis XVIII, the younger brother of the last French king. He would pretend for a time that the Revolution didn't occur, much to the chagrin of the French people. Yet it wasn't an absolute Monarchy; the government in place would be adapted into a Constitutional Monarchy to restrict the King and keep some semblance of structure that Napoleon and the Revolution created in France. Yet this wasn't enough, people yearned for Napoleon, the Napoleon that didn't care for rank but valor.

On the first of March, Napoleon would return from Elba. In a whirlwind march from the South of France to Paris, he would draw soldiers to him as they were sent in to chain him or as they heard of his return. The French army was literally rebuilt around him as he marched for Paris to an empty throne which Louis XVIII fled in fear. Right away Napoleon would work to restore his reign but more limited, he would accept a Constitutional Empire that would be sorted out once France was safe, and he set his eyes on the destruction of a Observational Army in the Netherlands under the command of Arthur Wellesley. Waterloo would come on July 18th, 1815; Napoleon would meet the would be title of a song by ABBA, and retreat once more into a permanent exile.

After the defeat of Napoleon, France (as well as Europe in general) would become oppressive in an attempt to stop any revolutionary ideologies from spreading. So there wouldn't be much of note for France to do during the reign of Louis XVIII. It wouldn't be until 1825 when Charles X becomes the last Bourbon that things start going poorly for the monarchy. The French parliament would start to see election results that would work against the Royalists in 1827 and 1829, causing many people to get angry when Charles X would dismiss several Prime Ministers simply because he didn't like them. This would result in the French Chambers of Deputies (somewhat similar to the American House of Representatives) eventually forcing him out after three days of protests, armed revolts, and effectively a new revolution. This Revolution of 1830 would be led not by the nobility like the early years of the French Revolution nor by the common people like the middle years of the French Revolution but by the middle class whom were yet again being barred from serving in government (particularly the Chamber). This middle class Revolution would place the "citizen king" of Louis-Philippe of the House of Orleans, the son of the Duc de Orleans (Philippe Egalite). Oddly enough it would be Charles X that would push for the invasion of Algeria to secure French Mediterranean trade, giving Louis-Philippe an Empire.

Louis-Philippe wouldn't be wholly popular as he would slowly turn from his liberal past to a more pro-monarch and conservative political base. This is beautifully shown in the June Rebellion shown in Hugo's Les Miserables of 1832. However France would turn against him in 1848 after a bad economic crisis the year before, throwing him out and enacting another Republic.

Thus enters the next Napoleon. Napoleon III (Louis-Napoleon Bonaparte) was the son of Napoleon's younger brother Louis Napoleon and Napoleon's daughter-in-law Hortense de Beauharnais. He would grow up being taught outside of France (due to exile placed by the fearful Bourbons) and became a devout Republican. He would return to Paris in time to leverage his name during the Presidental election of 1848, using latent Bonapartism in his favor as well as distaste for extreme Republicanism and Royalism that was prevalent in France (as you would hate one or the other). He was elected and slowly pushed against the Republican left until he outright enacted a proper coup in 1851 to establish himself as Emperor. He would enact a plebicite to establish his rule, which he handily won over but unlike his revered uncle, he would be dogged with questions over it's legitimacy.

And thus we return yet again to a French Empire. Napoleon III would punish republicanism in France, exiling many people (including Victor Hugo) to ensure that France would stay conservative. Rather than finding a medium between the right and left as his uncle would, Napoleon III pushed to the right. It wouldn't be until 1870 when Napoleon pushed France to war against the belligerent Prussia that the Second French Empire would fall.