Short answers, pulled from the story.
Maximilian I of Mexico was an Austrian archduke born on the 6th of July 1832 in Vienna's Schönbrunn Palace, and the younger brother of Emperor Franz Joseph I of Austria. He served as emperor of the Second Mexican Empire from the 10th of April 1864 until his execution by the Mexican Republic on the 19th of June 1867.
Mexican conservative monarchists sought a European royal to legitimize a Mexican throne, and Maximilian's membership in the House of Habsburg gave him historical associations with the Viceroyalty of New Spain. He was considered a capable administrator based on his time as viceroy of Lombardy-Venetia, and he was unlikely to inherit a European throne due to his elder brother Franz Joseph's position as emperor of Austria. Napoleon III brokered his invitation.
Maximilian was executed by a Republican firing squad on the Cerro de las Campanas in Querétaro at 6:40 in the morning on the 19th of June 1867. He was shot alongside his generals Miguel Miramón and Tomás Mejía Camacho after a trial in the Teatro Iturbide of Querétaro that returned a guilty verdict after a single day.
Maximilian's last words were: "I forgive everyone, and I ask everyone to forgive me. May my blood which is about to be spilled end the bloodshed which has been experienced in my new motherland. Long live Mexico! Long live its independence!" He spoke only in Spanish and gave each of his executioners a gold coin.
Maximilian upheld the liberal reforms of the Constitution of 1857, including freedom of worship and the sale of Church property, rather than reversing them as conservatives had expected. When the papal nuncio Pier Francesco Meglia demanded in December 1864 that Maximilian return Church property and reinstate Catholicism as the sole religion, Maximilian refused, severing his relationship with the Church and the conservative base that had brought him to power.
Carlota traveled to Europe in 1866 to plead for continued support for the Mexican Empire, but her mission failed. After the failure of her appeal, she became increasingly mentally unstable. She spent the rest of her life in seclusion in Belgium and lived until 1927.