FFE IFCE DRJSCS
In UK you have the BHS (British Horse Society: deals with Welfare and Care of the horse, Safety and Accidents, Access and Bridleways, and Training of Instructors grooms etc. Then each discipline has its own body running the sport's competitons: BSJA (British Show Jumping Association), BD (British Dressage), BE (British Eventing), BDS (British Driving Society), BEV (British Equestrian Vaulting) etc. etc. There are also a lot of competitions organised locally by different clubs or associations; these are called unaffiliated competitions, as you do not need to affiliate to any of the bodies above. The standards therefore vary, and they are usually of a lower level of difficulty.
In France it is quite different. Only 3 bodies:
The FFE (Fédération Française d'Equitation), which encompasses all the disciplines (so like The BHS used to do pre splitting into BSJA, BE, BD, BDS, BEV, etc. etc.). Their website is extremely complete. FFE club SIF is everything at equivalent of unaffilated level in UK, except that in France it is affiliated to the FFE. FFE Compet is equivalent to affliated level. You need a minimum level of riding to compete at that level (Galop 7). If you have competed affiliated in UK, you can get your body to write a letter saying that you did compete at that level, and the FFE will grant you an automatic Galop 7.
All is run by the FFE and its regional delegations (CRE: Comité Régional d'Equitation). FFE website: https://www.ffe.com/ The FFE also delivers "labels" (quality standards) for riding schools.
The IFCE (Institut Français du cheval et de l'Equitation). In 2010 it grouped together the Haras Nationaux (concerned with the welfare of the horse, registering every horse on French soil, and keeping up breeds and studbooks) and the ENE (Ecole Nationale d'Equitation; concerned with the Elite, training instructors, and the Cadre Noir). ENE is in Saumur - well worth visiting.
DRJSCS (Direction Régionale de la Jeunesse et des Sports et de la Cohésion Sociale), formerly known as Jeunesse et Sports. In France you have a ministry of youth and sports. They regulate the French Sports qualifications for teaching, riding schools have to be declared with them, they deliver a "carte professionelle" allowing you to teach (renewable every 5 years).