The Académie française rules on French and is routinely ignored, which is itself a reliable feature of the language.
Forty people, and no actual power

What the Académie actually is
There are forty of them. Forty Immortels — that is the official term, chosen without irony in 1635 when Cardinal Richelieu founded the institution under Louis XIII — and their charge is to write and maintain the dictionary of the French language. They sit under the gilded dome of the Institut de France on the left bank of the Seine, they wear embroidered green coats on ceremonial occasions, and they issue rulings on what French words are permissible, correct, or condemned. Their dictionary, the Dictionnaire de l'Académie française, has been in continuous production since that founding year, with the ninth edition still unfinished nearly four decades into the effort.
None of their rulings carry legal force. Not one. The Académie advises; France does as it pleases.

This is worth understanding clearly, because newcomers often assume the opposite. The institution sounds authoritative — it is ancient, it is state-funded, its members include writers of real distinction — and so when it declares that courriel is the correct French for email rather than the English loanword, it sounds like a decision rather than a suggestion. In practice, everyone around you says un email, and the Académie's preferred term quietly inhabits official documents and government websites, where it is dutifully ignored by the people who upload them.
The gap between ruling and usage
The Académie's relationship with the French public is one of fond, slightly theatrical disregard. France has strong feelings about its language — genuinely strong, not merely performed — and yet those feelings do not translate into compliance with the forty Immortels. They translate instead into heated conversations about which Anglicism is an outrage and which one is fine now. The distinction is made entirely without the Académie's input.
Some of its battles have become famous enough to function as shared jokes. The war on le week-end was lost before it properly began; le parking, le shopping, le hamburger followed. The Académie's 2017 ruling against the inclusive writing device known as l'écriture inclusive — which uses a mid-point to merge masculine and feminine endings, as in les étudiant·e·s — generated substantial press, a certain amount of outrage, a counter-outrage, and no measurable change to the practice among those who had adopted it. French universities and municipalities that wanted to use it continued to do so.

What the Académie does genuinely influence is the written register of institutions: ministerial communications, formal correspondence, the language of the préfecture. Here the weight of convention and the force of a shared, codified standard matter practically. If you are reading an official document and encounter a slightly stiff or archaic construction, there is a good chance it reflects Académie usage preserved in bureaucratic amber. The gap between that formal register and the French you hear at the market or on the Métro is wide enough to be occasionally disorienting for a learner.
The Immortels themselves are elected for life by sitting members, which produces a body that tends to reflect a certain kind of literary eminence rather than the actual spread of French as it is spoken and written — in France, in Québec, in West Africa, in the French Caribbean. The institution has occasionally recognised this. Marguerite Yourcenar became the first woman elected, in 1980, after three and a half centuries of male-only membership. The door opened slowly. As of the mid-2020s women remain a minority among the forty seats, though the proportion has grown.
Living with this institution means accepting a productive tension: the language has a guardian, the guardian's authority is limited, and French speakers are entirely at ease with the contradiction. The Académie endures not because it commands but because it anchors. Even when you ignore a referee, it helps to know one exists.