The ITF - International Tennis Federeration, formerly the ILTF - International Lawn Tennis Federeration since 1913, is the overall Governing body of World Tennis, wheelchair tennis, etc. It is responsible for the Rules of Tennis, for regulating equipment, surfaces, and administering tennis' Anti-Doping program, Anti-Corruption program, among other things.
As @RocketMan70; mentions, ITF has oversight over the majors, is responsible for the Grand Slam Rulebook, but permits each major's national tennis governing bodies (TA - Tennis Australia, FFT - French Federeration of Tennis, LTA (Lawn Tennis Association, USTA - United Statest Tennis Association) to run their own show within the rules. The current majors are the: Australian Open (AO), French Open at Roland Garros (RG), The Championships, Wimbledon (WC), the oldest and most prestigious tournament in the world, and the US Open (USO). The ITF also administers Davis Cup, Fed Cup, Olympics, and the various lower tours and leagues (Challengers, Futures, etc). The ATP (Association of Tennis Professionals) and WTA (Women's Tennis Association) are responsible for administering the the professional main men's and women's tours, respectively.
The ITF, ATP and WTA haven't always functioned well together. There have been a few disputes among them over the years with each trying to exercise their power. For example, currently the ATP refuses to grant ATP ranking points for Davis Cup and the Olympics due to ITF's lack of support. As usual this dispute was mostly about money. The ITF refused to compensate the minor ATP tournaments when the Olympics were played during their dates. The ITF said it didn't care if points were awarded or not, as they felt the players should play the Olympics, Davis Cup and Fed Cup for the honor of their country, not for prize money and points.
One would think that the ITF would have some authority over the French Federeration of Tennis and its unilateral decision to move their dates, but it appears that they have chosen for the time being to say nothing. It probably should come as no surprise, since the move comes at the expense of a major overrunning 18 ATP (including Laver Cup) and WTA tournaments. So the ITF are probably disinclined to act against the FFT. These internal squabbles probably don't halp the appearance of Tennis as a unified organized body. On the other hand, the competition among them probably encourages future innovation to enhance the sport, rather than letting it become stale. We shall see.