Wikipedia is your friend. Read the CNRS wikipage, and look at CNRS official website.
CNRS is the largest public pure research organization in France, with a staff exceeding 31000 persons, located in many geographical areas in France; many CNRS teams share office buildings with e.g. some French universities; several French research teams (UMR) have mixed staff, some funded by a University, and some other funded by CNRS, perhaps even working in the same office room on similar research. Getting employed by CNRS is really very hard, and there is a lot of competition. BTW, CNRS researchers are not very well paid, and most of them are not only very competent, but generally passionate about their research. They are not (contractually) tenured members of some University, even if most of them do teach a few courses somewhere (in addition of their researcher work), either at some University or at some Grande Ecole.
It's supposedly some unknown research center somewhere in Europe
CNRS is the largest basic research agency in Europe (mostly funded by the French public State budget). It is French (even if of course it gets some research grants from outside, e.g. from French ANR or European Commission's H2020...)
CNRS researchers are working in all kind of science and research (e.g. biology, history, computer science, physics, sociology, paleontology, chemistry, mathematics, etc etc etc ....). Most of them have at least a PhD (doctorat d'université), usually even their HdR (habilitation à diriger des recherches). Even if CNRS researchers are French civil servants, they are not all French citizens (even if most of them are).
There are some other public (French state owned) organizations in France doing research, e.g. INRIA, INRA, CEA, INSERM, (and dozen of other smaller research institutions) etc... Unlike CNRS, these other organizations are usually dedicated to some specific science or technological domain, and usually work on applied research.
Online material on the cnrs isn't helpful at all
Why do you say that? The Overview page of CNRS website is quite informative (and looks quite objective to my French citizen eyes)! And the CNRS wikipage gives a complementary look.
French economy is not used to fund research (in the sense that private corporations in France spend much less money in funding research, notably in public labs, than their counterpart in the USA or even in Germany). So the funding of French research works differently from North American country (and is lower, in terms of percentage of GDP).
BTW, I don't understand well what the NSF exactly is in the USA. My perception is that it mixes the role of CNRS and of ANR in France.