On 6 October, 1793 (15 Vendémiaire, An II), the Convention decided to create a new calendar for the new Republic, fixing the start date as the day when that Republic was proclaimed, namely the autumn equinox, 22 September, 1792.
As for the seven-day week, it was replaced by a ten-day cycle called a 'décade': day names were changed to primidi (oneday), duodi (twoday), tridi (threeday), quartidi (fourday etc.), quintidi, sextidi, septidi, octidi, nonidi and décadi. Months were made up of three decades, and the year ended after Fructidor with 5 supplementary days (Jours supplémentaires); and a 6th 'Jour de la Révolution' for leap years.
As for the months, new names for them were invented by the poet Philippe François Nazaire Fabre, known as Fabre d'Eglantine (1750-1794). He took as his inspiration the seasons and the events in Nature associated with them: the Republican year began with the month of Vendémiaire (from the Latin 'vindemia', grape harvest) (22 September to 21 October). This was followed by the months of Brumaire (from the French 'brume', fog) and Frimaire (from the French 'frimas', hoarfrost); then Nivôse (from the Latin 'nivosus', snowy), Pluviôse (from the Latin 'pluviosus', rainy), and Ventôse (from the Latin 'ventosus', windy); following that, Germinal (from the Latin 'germen, germinis' a bud), Floréal (from the Latin 'floreus', flowery) and Prairial (from the French 'prairie', meadow); and finally Messidor (from the Latin 'messis', corn harvest and the Greek 'doron', gift), Thermidor (from the Greek 'thermon' heat and the Greek 'doron' gift) et Fructidor (from the Latin 'fructus', fruit and the Greek 'doron', gift).
Names were then given to every day of the year (PDF). These were chosen from the names of flowers, trees, plants, animals and farm tools.
Aware of the unwieldy nature of a calendar whose first day in the year (the irregular autumn equinox) was never the same day, and in a conscious attempt to detach the newly founded Empire from the Revolution and to set it within the context of the whole of French history (right back to Charlemagne), Napoleon I abolished the calendar by a Décret impérial of 9 September 1805 (22 Fructidor, An XIII). With the Gregorian calendar beginning again on 1 January, 1806, the Republican calendar had lasted 13 years. It was however to have an Indian Summer during the Commune... from 6 to 23 may, 1871.