historians actually split the revolution into two simultaneously occurring revolutions, the radical revolution and the liberal revolution. The radical revolution was run by people such as the ever present hordes of paris (sans culottes) (think heads on pikes, slicing up bodies of royalists to completely and symbolically destroy them, rioting, and a complete disdain for royals and the first and second estates) Jacobins, and eventually the committee of public safety.
When Jacobin powers in the assembly won the vote to execute the king, they rose in power over the Girondins through popular opinion and by simply removing many girondins from the assembly. This is when the radical revolution really got into full swing with Robespierre at the helm. this was the time where you could be guillotined for basically any sort of suspicion and the people of Paris especially were under a reign of terror.
Radicals thought that France needed to be purged of any sort of social hierarchy and be disconnected from the church completely. They mistrusted any executive power. Radicals in Paris believed they should be able to form an army and go out into the country and take food by force from the farmers. (one of the biggest problems in paris were food prices, and general lack of food) They believed that they, as the working class, were constantly being taken advantage of by richer people and called for ridiculously exorbitant and arbitrary taxes to be levied on these poeple.
Im taking this class right now so let me know if I can help more, but thats all i got off the op of my head