Short answers, pulled from the story.
The Second French intervention in Mexico began after Mexican president Benito Juárez placed a moratorium on foreign debt payments and expelled all Spanish diplomats in July 1861. This action triggered a response from Spain, France, and Great Britain who agreed to the Convention of London to ensure debt repayments would be forthcoming.
Mexican forces commanded by Ignacio Zaragoza and Porfirio Díaz won a major victory against the French at the Battle of Puebla on the 5th of May 1862. A Mexican force of about 5,000 men confronted 6,000 French troops as they tried to climb steep terrain towards the city.
On July 11, the Assembly resolved that Mexico was to be a constitutional monarchy and invited Austrian archduke Ferdinand Maximilian to accept the Mexican throne. Maximilian and his wife Charlotte of Belgium arrived in Veracruz in the summer of 1864 and were later crowned in the Mexico City Metropolitan Cathedral.
The United States never recognized the government of Maximilian but could not aid the Republican cause until its own civil war ended in April 1865. Secretary of State William H. Seward stated that it did not accord with U.S. policy to acknowledge a monarchical government erected on the ruins of any republican government in America under the auspices of any European power.
Maximilian was executed on the 19th of June 1867 along with Generals Miguel Miramón and Tomás Mejía on the Cerro de las Campanas outside Querétaro. Mexico City surrendered the day after Maximilian's execution following a court-martial where Juárez refused to commute the sentence despite pleas from many European monarchs and other prominent figures.