It's important to understand that unlike the American revolution, where colonists sought independence from the British Crown, the French repudiated existing society, often referred to as the ancien regime (old order), and sought to replace it with new political, social, and cultural structures. But, unlike their American counterparts, French revolutionaries lacked experience with self-government.
Serious fiscal problems put France on the road to revolution. In the 1780's approximately half of the French royal government revenue went to pay off war debts - some of them arising from French support for colonists in the war of American independence-and an additional quarter went to French armed forces. King Louis XVI sought to raise taxes on the peasantry, and protests against this caused him to call the Estates General, an assembly that represented the entire French population through groups known as estates. In the ancien regime there were three estates, the clergy, the nobles and the peasantry. The third estate had as many delegates as the two others combined, but numerical superiority offered no advantage when voting on issues, because voting took place by each estate - one vote each - not by individuals.
On June 17, 1789, representatives of the third estate took the dramatic step of seceding from the Estates General and proclaiming themselves to be the "National Assembly". Three days later, meeting in an indoor tennis court, members of the new assembly swore not to disband until they had provided France with a new constitution. On July 14, 1789, a group of Parisians, fearing the king south to undo events of the past weeks, stormed the Bastille, a royal jail and arsenal, in search of weapons. The military garrison protecting the Bastille surrendered to the crowd but only after killing many of the attackers. To vent their rage, members of the crowd hacked the defenders to death. One assailant even used his pocketknife to sever the garrison commander's head, which the victorious crowd mounted on a pike and paraded around the streets of Paris. News of the event soon spread, sparking insurrections in cities throughout France.
Between 1789 and 1791 the National Assembly reconfigured French society. Taking :liberty, equality and fraternity" as their goals, and basing actions of the document they had written earlier, Declaration of the Rights of Man and the Citizen, the Assembly abolished the old social order along with many fees and labor services that peasants owed to landlords. They also abolished the first estate, declaring the clergy as civilians and requiring them to take an oath of loyalty. France became a constitutional monarchy in which men of property-about half of the adult male population-had the right to vote in elections. Thus far, the French revolution represented an effort to put Enlightenment values into practice.
The revolution soon however took a radical turn, this is the meat of what you were asking about. Efforts by the French nobility to mobilize foreign powers in support of the king gave the Assembly the pretext to declare war on Austria and Prussia. Adding to the burden, leaders declared war the following year on Spain, Britain and the Dutch. The convention rallied the French by instituting the levee en masse (mass levy) or universal conscription that drafted people and resources. The convention also rooted out enemies at home, making frequent use of the guillotine. In 1783, King Louis XVI and his wife Marie Antoinette were killed.
Chaos reached its peak under Maximilien Robespierre and the Jacobin party dominating the convention between 1793 and 1794. A lawyer by training, Robespierre had emerged during the revolution as a ruthless but popular radical known as “the Incorruptible” and he dominated the Committee of Public Safety, the executive authority of the Republic. THe Jacobins believed passionately that France needed complete restructuring, and they unleashed a campaign of terror to promote their agenda. Th sought to eliminate the influence of christianity in French society by closing churches and forcing priests to take wives. The promoted a new “cult of reason” as a secular alternative to Christianity. They reorganized the calendar, keeping onths of thirty days but replacing seven day weeks with ten day ones that recognized no day of religious observance. The Jacobins also proclaimed the inauguration of a new historical era with the Year I, which began with the declaration of the First Republic on September 22, 1792. They encouraged working class citizens to display their revolutionary zeal by wearing working-class clothes. They even granted more agency to women, by permitting them to inherit property and divorce their husbands. Finally the Jacobins made great use of the guillotine. Between the summers of 1793 and 1794 they executed about forty thousand people and imprisoned three hundred thousand suspected enemies of the revolution.
Many victims of the reign of terror were actually fellow radicals who fell out of favor with Robespierre and the Jacobins. The instability of the leadership eventually undermined confidence in the regime itself. In July 1794 the convention arrested Robespierre and his allies, convicted them of tyranny, and sent them off to the guillotine. A new institution was set up known as the Directory was set up and ruled for a few more years until Napoleon took over.
If you’re interested, I’d highly recommend reading some Robespierre and what he had to say, as he by far caused the most violence in the revolution. Here is a link to one of his speeches.