The monarchy was restored whilst napoleon was in exile in Elba. Napoleon returned from exile and gathered an army. He was then defeated at Waterloo (although by that time he had arguably already lost).

You mean before the 2nd defeat of Napoleon?

After defeats in Germany including his first major tactical defeat at Leipzig the Allies crossed into France, and from Spain across the Pyrenees from mid 1813 into 1814 Napoleon was forced to abdicate by his chief Marshals when confronted. This final war saw the return of Austria to the fighting, the advance of Russian forces across Europe, Prussia returned to the fore and opposing Napoleon and uprising across Germany, while Wellington with the British-Portuguese-Spanish forces near total victory in Spain and advancing into Southern France.

He had won several victories during the years fighting, but was severely outnumbered and the quality of his forces had eroded as well. While he stole some victories in his typical aggressive bold strokes such as the 6 Days Battles, he had neither the army, nor the space to maneuver for an eventual victory. When Paris fell at the end of March that was essentially the breaking point. The Allies had announced that while they would no longer negotiate with Napoleon, they would talk with a new government, either provisional or a restored monarchy, that they would respect her prewar borders, and were not looking to destroy the nation. An a Senate set up in Paris voted out Bonaparte in the first few days of April and accepted those terms. Napoleon tried an end around by abdicating in favor of his young son and a regency but this was rebuffed.

For a host of reasons in this context and for reasons both selfish and selfless those senior officers still with the Emperor then at Fontainebleau pressured him to accept exile to the island of Elba vs continuing a lost war. Centered around Marshall Ney the main thrust was that Napoleon could order whatever he liked but without them to carry out the orders the army would collapse anyway into competing camps of loyalty and he was still done for.

So the treaty is signed tot hat effect and with a small honor guard Napoleon goes into exile in the Mediterranean and Louis XVIII gets the throne. However after some initially positive signs tensions soon rose and Napoleon saw his chance to reclaim the throne, win a handful of victories, and if nothing else restore some territorial gains with the help of an army made of up veterans, new levies, and returned POWs. While perhaps playing the other powers off against each other who were focused on the Congress of Vienna.

So began the 100 Days on February 26 1815 when he left the island and really got started on March 20th when he entered Paris at the head of an army grown from defections. The Royal family by then had fled to Ghent for protection. Napoleon's attempts to negotiate peace, rebuild his forces, and secure his rule would end up for naught when he was defeated by a British-Dutch-Prussian force in the Waterloo campaign and captured in flight. This time there would be no messing around and he was exiled to St. Helena. And then Louis would be able to return to ruling in Paris, thought this time was forced to agree to a treaty which forced him to pay indemnities to the Allied Powers, fund an occupation force, and lost some territory.

So yes it is ok to say the start of the Bourbon restoration was April 1814. Because Louis did return to rule though with some negotiations still incomplete but Napoleon was absolutely out of power by then and would be for most of a year. But with the one caveat being that for some months he would be forced to rule in exile though he never gave up his crown and there were competing French monarchies in the Bourbons and Bonaparte's for a few months in 1815.