The French Revolution was a deeply transformative (as well as destabilizing) period in history, and its repercussions continued to be felt across the nineteenth century. It's also deeply difficult to pin down because of how it evolved across time, going through several different stages, each of which had its own particular accomplishments as well as failures.

I would say that, at its core, the French Revolution envisioned a radically new understanding of the relationship between a people and their country and of the notion of where sovereignty originates. Under Absolutist theory, sovereignty was ultimately viewed as being held within the monarchy itself, but the French Revolution would break with that traditional understanding. Instead, the Revolution would hold that sovereignty must ultimately reside with the people and nation of France. To simplify this argument, one might say that the French Revolution entailed the transformation of subjects into citizens and was key in unleashing Nationalism on Europe.

The first stage of the French Revolution would include the abolition of the remnants of Feudalism and the installation of the Declaration of the Rights of Man. In addition, it would install a new system for administration, by which France would be divided into departments. To quote historian Jeremy Popkin,

These units, of approximately equal size . . . were intended to rationalize France's administrative system and also to break down provincial loyalties that might conflict with loyalty to the nation as a whole. The maps of the many overlapping court districts, religious dioceses, and other administrative divisions of the Old Regime were redrawn to correspond to departmental boundaries, which made citizens' relations with the government much simpler. With minor modifications, the departments created in 1789 have survived to the present day. (Popkin, A Short History of the French Revolution (3rd ed.), Upper Saddle River: Prentice Hall, 2002, 48)

Of course, this was only the beginning of a much longer history which would have its own twists and turns. With the abolition of the monarchy (and the execution of Louis XVI), the Revolution would enter a still more radical phase, infamous for the Reign of Terror. By this point, the Revolution would come to represent a complete break with the pre-Revolutionary past, with the old Gregorian Calendar being abolished and replaced with a new Revolutionary Calendar, which would mark the creation of the Republic as the New Year 1. With the fall of Robespierre, we will see the Thermidorian Reaction and the creation of the Directory. Finally, there's the specter of Napoleon to keep in mind, as well as the establishment of first the Consulate and then the Empire.

Some of the burdens thrust upon the peasantry were alleviated by the French Revolution. Taxation was less harsh and for a while the people had a reprieve from the oppression of the aristocratic classes, which made social class so fixed in the country. The National Assembly also declared that all men were born equal and deserved the same human rights universally, though this seemed to be observed more as an abstract theory than as a concrete reality.

The Napoleonic Code was another major achievement of the Revolution, creating a code that many a capitalistic society would function by for centuries to come, not only in France, but abroad.

The separation of church and state was a major achievement of the Revolution as well.

The French Revolution that took place from 1789 to 1799 achieved the following;

• Led to the abolition of the monarchy at the time headed by Louis XVI who was sentenced to death for treason and other crimes against the state.
• Laid the foundation for the establishment of the French Republic even though this was actualized much later.
• The feudal land rights of the lords were eradicated, and the church land was repossessed by the government. Peasants finally got access to this land.
• The taxation structure was amended to ease the tax burden on peasants and members of the third estate.
• Adopted the Declaration of the Rights of Man and of the Citizens which enabled the citizens, especially the common populace, experience the benefits of freedoms and rights albeit briefly.
• A jury system was introduced for the purpose of handling criminal cases.
• Gave rise to a new way of thinking that spread to greater Europe and resulted in democratic revolutions, as well as nationalism and romanticism.

Obviously, the French Revolution went in directions that many of its liberal (in the 18th century sense) advocates did not envision. So its achievements always have to be weighed against the human costs of the Revolution, and some achievements had decidedly mixed results. For the purposes of this list, I will also include achievements that were either begun or consolidated under Napoleon, which some scholars consider part of the Revolution:

• the legal abolition of the nobility, including its feudal privileges, by the National Assembly in 1789.
• the Declaration of the Rights of Man and of the Citizen, considered one of the world's most important statements of universal human rights, issued just after the abolition of the nobility.
• the reogranization of France's inefficient governing system into departments and communes.
• the temporary establishment of equal rights for blacks in France's colonial possessions. An uprising in Saint-Domingue, which ultimately led to the Haitian Revolution, would decree freedom for all slaves, though the French tried to reestablish it under Napoleon.
• Legal equality for Protestants and Jews, established in 1791.
• The establishment of the metric system in 1793.
• Formal separation of church and state in 1794.
• National system of schools known as lycées established in 1803. 
• System of French civil law known as the Code Napoleon established in 1804.

Another achievement of the Revolution, it could be argued, was the stimulation of liberal nationalism throughout Europe. Obviously, this spirit was most successfully manifested, albeit very temporarily in Haiti, but it also inspired people on the continent, as well as in Latin America.