The French Revolution brought us two things, a way of organizing the state and a word for describing it, "bureaucracy". Of course, government has always taken place at different levels. Some terms used to this day, like "county", are rooted in feudalism. Literally, a "county" is the domain of a count. Other terms for places are related to the organization of the church, for example, "parish" in Louisiana. Modern states have also added many other terms to an already complex vocabulary for the various divisions of the state.
For a genealogist, the many different levels of government can present difficulties. These words are in a sense untranslatable. The meaning of a term cannot be understood without reference to the particularities of the organization of a country. For an American, the term "state" has a significance which it will not have for other speakers of English. Thus when approaching a new country, it is important not only to translate the names of the various units of organization but understand the system on its own terms.
Making your way in France
The different levels of organization in post-Revolution France present their own particular difficulties for genealogists. The largest unit is the "région". The names of the regions will probably be the most familiar to you. They include "Alsace", "Aquitaine", "Brittany" and other famous names. If you have a family tradition concerning where your family is from in France, the names of the regions may provide important clues. However, you need to be careful because the names of regions may not always correspond to the more traditional use of these terms. The city of Nantes was traditionally a part of Brittany, but now belongs to the Pays de la Loire region.
The next level down is the "départment". This is the second most important level for genealogists. Departments maintain their own archives, many of which can be easily accessed online. In my own research, I have extensively used the websites for Côtes-d'Armor and Morbihan in Brittany and Loire-Atlantique in Pays de la Loire.
Below the department level are the "arrondissement" and the "canton". You can usually just ignore these levels.
Finally, you come to the "commune". The commune is the most important level for genealogists. This is the level at which most records are collected. So, if you are looking for a marriage, death or birth record, you will need to figure out the commune name.
So, to recap, you need to identify both the commune and department in which your ancestor lived. The department will tell you where you need to look for an archive with the correct records. Once, you have found the right place, you will need to find the records for the commune. Only knowing the birthdate and birthplace of my great-grandmother, I have been able to trace back her ancestry to before the French Revolution across three departments.