Short answers, pulled from the story.
The French Republican calendar was a calendar created during the French Revolution, used by the French government from late 1793 to 1805, and briefly by the Paris Commune for 18 days in 1871. It replaced the Gregorian calendar with twelve 30-day months, three 10-day weeks per month, and five or six extra days at year's end to complete the solar year.
The poet, actor, and playwright Fabre d'Églantine invented the names of the months, with the help of a commission that included the politician Gilbert Romme, the astronomer Jérôme Lalande, the mathematician Joseph-Louis Lagrange, and the gardener André Thouin. The month names were based on nature, principally the weather and agriculture around Paris.
Napoleon abolished the calendar by a decree signed on the 22nd of Fructidor An XIII, citing a report that identified two fundamental flaws: leap years depended on astronomical observation rather than fixed arithmetic, and the calendar's seasonal structure was tied to metropolitan France, making it incompatible with the rest of Europe and the Americas. The unpopular 10-day week, which gave workers fewer rest days than the Gregorian seven-day week, had already been suppressed three years earlier.
The tenth day of each ten-day cycle, called décadi, replaced Sunday as the official day of rest and festivity. Décadis were the festival days for a succession of new religions promoted by the Revolution, including the Cult of Reason, the Cult of the Supreme Being, the Decadary Cult, and Theophilanthropy. A law of 13 Fructidor Year VI required that marriages could only be celebrated on décadis.
The rural calendar, introduced by Fabre d'Églantine, gave every day of the year a unique name drawn from agriculture and nature, replacing the Church's calendar of saints. Every décadi was named after an agricultural tool, every quintidi after a common animal, and the remaining days after plants, grains, flowers, and fruits, except during the winter month Nivôse, when the non-tool and non-animal days were named after minerals.
The Coup of 18 Brumaire on the 9th of November 1799 brought Napoleon to power and gave its name to Karl Marx's 1852 essay The Eighteenth Brumaire of Louis Bonaparte. The 9th of Thermidor An II, the 27th of July 1794, was the day the Convention turned against Robespierre, making Thermidor a political term for a revolution turning on its own radicals. Émile Zola named his novel Germinal after the calendar's spring month, and Lobster Thermidor took its name from an 1891 play set during the Revolution.