Short answers, pulled from the story.
The Second Federal Republic of Mexico was a period of Mexican history lasting from 1846 to 1863, beginning with the restoration of the Constitution of 1824 during the Mexican-American War and ending when the Second French Intervention led to the proclamation of the Second Mexican Empire in July 1863.
La Reforma was a series of sweeping constitutional reforms initiated after the liberal Plan of Ayutla overthrew Santa Anna in 1855. Its landmark achievement was the Constitution of 1857, which separated church and state, nationalized Catholic Church lands under the Ley Lerdo, abolished special clerical and military courts, and declared civil rights including freedom of speech and freedom of the press.
France used Mexico's 1861 moratorium on foreign debt payments as a pretext to invade, but Napoleon III's actual goal was to overthrow the Mexican Republic and establish a monarchical client state. The idea had been proposed to Napoleon by two Mexican monarchist expatriates, Jose Manuel Hidalgo y Esnaurrizar and Jose Maria Gutierrez de Estrada. Britain and Spain withdrew from the joint expedition once they understood France's true intentions.
On the 5th of May 1862, Mexican troops repulsed French forces at the Battle of Puebla, delaying the French advance on Mexico City by over a year. French reinforcements eventually arrived, and by June 1863 the capital had fallen to French forces.
Benito Juarez was a liberal lawyer who served in the cabinet of Juan Alvarez and later became president of Mexico after President Comonfort resigned during the Reform War. Juarez led the Liberal government from Veracruz through the three-year Reform War, won the decisive Battle of Calpulalpan in December 1860, and then faced the French Intervention, evacuating Mexico City ahead of French forces in June 1863.
The Revolt of the Polkos was an uprising that broke out on the 27th of February 1847 when national guard battalions in Mexico City proclaimed against the government of Valentin Gomez Farias. The revolt was triggered by a proposal to seize fifteen million pesos from the Catholic Church. The name came from the young middle-class men who made up the militia units and were known for dancing the polka.