Short answers, pulled from the story.
The Second French intervention in Mexico was a military invasion of the Republic of Mexico by the French Empire of Napoleon III, lasting from 1861 to 1867. Napoleon III used the pretext of unpaid Mexican debts to justify the invasion, but his broader aim was to establish a French-backed monarchy in Mexico and limit the growing power of the United States in the Americas.
Maximilian was Austrian Archduke Ferdinand Maximilian, chosen by Napoleon III and a conservative Mexican Assembly of Notables to become Emperor of Mexico. The Assembly resolved on the 11th of July 1863 that Mexico would be a constitutional monarchy and invited Maximilian to the throne; he and his wife Charlotte of Belgium arrived in Veracruz in the summer of 1864 and were crowned in the Mexico City Metropolitan Cathedral.
Mexican forces commanded by Generals Ignacio Zaragoza and Porfirio Díaz defeated a French force of roughly 6,000 troops at Puebla on the 5th of May 1862, inflicting 476 French casualties against 227 Mexican losses. The victory, now commemorated as Cinco de Mayo, delayed the French advance on Mexico City by a year and strengthened President Juárez's political standing.
Napoleon III announced the withdrawal of French troops in January 1866 under combined pressure from rising U.S. diplomatic demands, the mounting cost of the occupation, and the growing military threat of Prussia on France's own European borders. The end of the American Civil War freed the United States to supply the Mexican Republicans with arms and money, making the French position in Mexico untenable.
On the 2nd of October 1865, Maximilian issued the Black Decree ordering that any combatant captured with arms would be executed without prisoner exchange, stating "Every individual of whatever rank, taken with arms in his hands, shall be put to death." The decree backfired by creating republican martyrs and deepening opposition to the empire.
The intervention ended on the 19th of June 1867 when Maximilian, General Miguel Miramón, and General Tomás Mejía were executed by firing squad on the Cerro de las Campanas outside Querétaro, after Republican forces breached the city's defenses when imperial Colonel Miguel López opened the gates in exchange for gold. Mexico City surrendered the day after the executions, restoring Juárez's republican government.