The fall of Robespierre after his arrest on 9 Thermidor an II (July 1794) ended the Terror and led to a more "civilized" stage of the Revolution. It meant also a resurgence in the royalist activity, and in 1795, the Convention was afraid that monarchists might win the next election. As a precaution, it adopted in August 1795 a decree, called "of the two thirds" stating that 500 of the 750 members of the future two assemblies will be current members of the Convention.*
* You think it had more to do with consolidating their power, don't you? I hate that you've become so cynical. And well done.
Those elections were scheduled for 20 Vendémiaire an IV (mid-october 1795), and Paris royalists were determined to make the Convention revoke the rule. On 11 Vendémiaire, seven Paris sections declared themselves as insurgents. Some of the National Guard joined them.
The Convention ordered Barras to take military measures and, on the field, Bonaparte became the de facto military leader. On 13 Vendémiaire (October 5), the insurgents (some mention 25,000 people) marched on the Convention, at the Tuileries. Bonaparte had put in place a number of cannons (procured by Joachim Murat) and, after receiving the order from Barras, told the troops to open fire. Grapeshot was very efficient on such urban terrain, causing around 300 deaths among the Royalists.
That action earned Bonaparte the nickname of "Général Vendémiaire" and is considered by some as the true beginning of his rise to power.**
** Oh yes - Napoleon earned his fame firing grapeshot at crowds of protesters. What a hero! He also earned several enemies - can't imagine why.