Short answers, pulled from the story.
Emperor of the French was the title of the monarchs of the First and Second French Empires, used to indicate that sovereignty belonged to the French people rather than to the territory of France. The older title King of France implied the monarch personally owned France as a possession. The new title framed the emperor as governing the nation by its consent.
Napoleon Bonaparte was proclaimed Emperor by the Senate on the 18th of May 1804. He crowned himself at the cathedral of Notre-Dame de Paris on the 2nd of December 1804, using the Crown of Napoleon.
The title Emperor of the French placed sovereign authority in the French people as a nation, preserving the appearance of the republic and signaling the end of the feudal system. It also reached back to the Roman title Imperator, a civic office associated with citizens rather than inherited power, framing Napoleon as Chief Magistrate and Commander-in-Chief elected by the nation.
Napoleon I reigned as Emperor from 1804, with his first period of rule interrupted by the Bourbon Restoration of 1814 and a brief exile on Elba. After escaping from Elba, he reigned for 111 days before his defeat at the Battle of Waterloo on the 22nd of June 1815. He died on the island of Saint Helena on the 5th of May 1821.
Napoleon III officially became Emperor of the French on the 2nd of December 1852, following a referendum held on the 7th of November 1852 that established the Second French Empire. The date was chosen because it fell exactly 48 years after his uncle Napoleon I's coronation as the first Emperor of the French.
Charles, Prince Napoléon, became head of the House of Bonaparte on the 3rd of May 1997 and is considered the primary claimant to the French throne. His claim is disputed by his own son, Jean-Christophe, Prince Napoléon, who was named as heir in his late grandfather's testament.