There is no supreme court in France, this is a gross mistranslation of "Court de Cassation", which is where you bring your case after you have lost your appeal, and is the last court where you can try to argue your point.
- Cour de Cassation, for civil matters
- Conseil d'État, for matters regarding the administration / the State
- Tribunal des Conflits: tasked with deciding who's right when the Cour de Cassation and the Conseil d'État disagree
- Conseil Constitutionnel: issues rulings about the constitutionality of laws, both new (before they become law) and existing ones (QPC)
This doesn't stop here however; there are two upper courts in the European Union, than can invalidate decisions issued by national courts:
- Court of Justice (in Luxembourg)
- Court of Human Rights (in Strasbourg)
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Edit: Don't you love the idea of "Tribunal des Conflits"? The original idea was that the State could not be brought to court, its decisions being made by "the people" who is the absolute sovereign.
Then France gradually accepted the idea that State's decision could be challenged, and created a whole different judicial system, the "justice administrative". It took a looong time: from 1800 to... 1980. A much simpler approach could have been to let people try their case against the State before the existing courts, but no... much better to build another system with its own rules, its own judges, etc.
An inevitable consequence of having two different systems is that they sometimes disagree. (Another reason why it would have been so much simpler to just have one system.) Since the two systems are sometimes at odds with one another, we created... a third system! This was in 1872, so quite early in the process.
This Tribunal des Conflits is a referee of sorts whose only job is to stop the fights between the two justice systems. I think that's great and tells a lot about the French way of solving problems: just add a new bureaucratic authority on top of all existing ones.
Le Conseil d'Etat est la juridiction administrative suprême.
Note that's it's not civil matters but matters related to government action (from say, basic rights to labor disputes for State employees or citizenship issues).
In any case, France has a civil law system; there cannot be a court as powerful as the Supreme Court of the US is. Viewing any of these institutions as similar to SCOTUS is bound to create a lot of confusion.
"The" supreme court, if one assumes a US-centric definition, comes with a lot of assumptions on the nature of law and the power structure of the various government branches. Which generally do not hold outside of the US and certainly not in France.
So yes, it is "a" supreme court, but that doesn't really help understanding, because it is not "the" supreme court.
Interestingly enough the last three presidents renounced their seats (I don't know why)