Short answers, pulled from the story.
The French Directory was the executive body established by the French Constitution of 1795, governing the First French Republic from the 26th of October 1795 until the 9th of November 1799. Executive power was held by a committee of five directors chosen by the legislature, with one director replaced by lot each year to prevent the concentration of power in a single person.
The French Directory was overthrown on 9-the 10th of November 1799 in the Coup of 18 Brumaire, engineered by Abbé Sieyès and carried out by Napoleon Bonaparte. The coup abolished the Directory and replaced it with the French Consulate, with Bonaparte as its leading figure.
The original five directors chosen in October 1795 were Paul Barras, Jean-François Rewbell, Louis-Marie de La Révellière-Lépeaux, Étienne-François Le Tourneur, and Lazare Carnot. Carnot replaced Abbé Sieyès, who was elected by the Council of Ancients but refused the position, saying it did not suit his interests or personality.
The Directory inherited an empty treasury, a collapsed paper currency (the assignat), soaring inflation, and a population that had largely stopped paying taxes. To stabilize finances, the Directory destroyed the assignat printing presses in a public ceremony at the Place Vendôme on the 19th of February 1796, imposed a forced loan of 600 million livres on wealthy citizens, minted 72 million silver coins from confiscated church property, and in 1797 declared bankruptcy on two-thirds of the national debt.
The Conspiracy of Equals was a plot organized by François-Noël Babeuf, who took the name Gracchus, to overthrow the Directory and establish a government based on equal sharing of wealth. Babeuf was arrested on the 10th of May 1796, tried in Vendôme, and guillotined on the 27th of May 1797 along with his principal co-conspirator Augustin Alexandre Darthé, after both men attempted but survived suicide.
The Directory established 29 short-lived sister republics in Italy, Switzerland, and the Netherlands. These included the Batavian Republic in the Netherlands, the Cisalpine Republic in Milan, the Ligurian Republic in Genoa, the Piedmontese Republic in Turin, the Roman Republic (proclaimed the 10th of February 1798 after French troops occupied Rome), and the Helvetic Republic in Switzerland (proclaimed the 12th of April 1798).