What are CNRS research units and how are they staffed

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.

CNRS research units, also called laboratories, are of several kinds:

• The 952 Unites Mixtes de Recherche ("Mixed Research Units", UMR) are run in association with another institution, for example a university, or another research agency such as Inserm, INRA... They are created by a contract between CNRS and the other institution for a five year duration (renewable). They are staffed by CNRS personnel and by the other institution's personnel. They are typically headed by a university professor or a CNRS directeur de recherche (research director, senior scientist), who presides over a laboratory council which has statutes, internal rules and regulations. Besides the researchers themselves, CNRS (and the other institution) also employs technical and engineering staff, as well as temporary staff (postdocs, PhD students) which are members of the laboratory.
• The 32 Unités Propres de Recherche ("Proper Research Units", UPR) are fully run by the CNRS. They are staffed by CNRS personnel. Other than that they are similar to UMRs.
• The 36 Unités Mixtes Internationales ("Mixed International Units") are international research units, created by a contract with a foreign institution. I don't know much about them but I expect that they are run similarly to a UMR.
• The 135 Unités de Service ("Service Units") are support units for other research units. An example would be a celestial observatory, who provides services for e.g. astrophysics research labs. It is also created by a contract between CNRS and another institution for a fixed duration, renewable.
• Finally, there are 126 public-private research structures, run in collaboration with private companies. Some of them are mixed research units (cf. above), but most (110) are simply common laboratories, which are created on a contract basis for 4 years.

Each of these is part of one of the ten national institutes of the CNRS (biology, chemistry, ecology and environment, human and social sciences, information and computer science, engineering and systems, mathematics, physics, nuclear and particle physics, universe sciences). Each unit is overseen by one of the 18 regional delegations.

CNRS researchers are hired on a national basis by the national committee for scientific research (CoNRS). More precisely, they are hired by one of the 41 sections of the CoNRS (which are sub-subdivisions of the 10 national institutes), each one composed of elected members and appointed members (by the ministry as proposed by CNRS). After being hired during the yearly national admission campaign, a researcher is posted in a research unit (of any kind), and may move to another after some years.

They are automatically tenured, like all French civil servants, and their missions, duties and responsibility are roughly equivalent to those of an associate professor (for junior scientist) or full professor (for senior scientists) — the main obvious difference being the lack of teaching.

More information can be found in Wikipedia, the website of the CNRS, or, if you are courageous, French law (here is the law defining the status of CNRS employees) – indeed, recall that CNRS is a public research agency, its employee are civil servants, and its mission and its organization are defined by law.

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The comparison with the NSF is a bit misleading. As far as I understand the NSF does not directly employ researchers but instead funds university to hire them through grants. This would be closer to what the National Research Agency (ANR) and the European Research Council (ERC) are doing.

As requested, I'm turning several of my comments into an answer. Please note I'm giving quite a superficial view - both because I don't know much more and because a detailed analysis would be too long for an answer.

To understand the CNRS, you really need to understand the French legal, political, and cultural context. France has very strong protections for permanent employees of any organization. It is essentially impossible to fire anyone. I vaguely remember reading somewhere that Pan Am, the airline that went bankrupt 20+ years ago, still to this day has employees staffing counters at Charles de Gaulle airport (with no customers) because it would not be worth the trouble to close down and fire them. I might remember wrong, but feels like a reasonable exaggerated version of the truth. It is also very hard to reassign the basic duties of an employee. If their original contract says their job is to do X, then the employer cannot start asking them to do Y instead.

In the US, most of the NSF and NIH budgets are used to indirectly employ researchers. The NSF/NIH contracts with universities, paying the university to conduct research. The university in turn uses the money to employ researchers and pay their salaries (as well as pay for their office space, staff support, et c). If the US government decides to reduce research funding, the university can fire the researchers, or at least reassign them to other duties such as teaching classes. (Note that employees of the US government do have civil service protections that make them harder to fire, so it would be harder to reduce or reallocate research funding if researchers were hired directly by the government.)

In the context of French employment law, no university (or any other entity) would be foolish enough to employ someone they can't fire (or even reassign their duties) when the funding for their position is temporary! Hence, the government is essentially forced to directly hire researchers (and assign them to work at universities) rather than giving funds to universities to hire researchers. (This is not quite right, because the ordinary French universities (i.e. excepting the Grandes Ecoles) are much closer to being part of the government than in the US, where even universities that receive government funding are usually quite independent.)