Quadrupedal movement (QM) is by definition any movement that uses hands and feet for locomotion. It includes all the fundamental activities: walking, running, jumping, balancing, climbing (which is mostly a specialized form of vertical QM), swimming, crawling, fighting (for instance in wrestling, but many other martial arts incorporate QMs), lifting and carrying, throwing, and all sorts of saving moves when falling.
So basically QM is more a fundamental attitude of the body. In fact, it is the first form of locomotion humans learn while growing up, and form a transitory type of motion before bipedalism (which is quite unusual if not unique in the animal world, by the way).
And yet, QM has been historically discarded in physical training as it "lowers Man to the level of beasts". It brought quite a few strong critiques on Georges Hébert's method and could not really be openly taught until the end of World War I, where Hébert demonstrated its vital importance for hiding from enemy fire while moving around. Nowadays, "going back to our natural roots" or "unleashing the inner monkey" are popular ideas, yet QM is still not widely taught as a general physical skill.
Perhaps the most important application of QM in sports is to protect oneself in the event of a fall. Most saving moves are indeed quadrupedal in essence, and so working on QM in general is important to build the strength and agility needed to fall without harm.
QM also leads naturally to some of the most athletic abilities in gymnasts, acrobats and now traceurs, such as flips, hand springs, and so on. Georges Hébert repeatedly mentioned his admiration of circus artists as most accomplished athletes and masters of the "transcendental QM".
This particular topic, too technically narrow and difficult for a general audience, is however not included in the book. Hébert's method was not aimed at training high-level athletes, but rather at describing the fundamental human movements that anyone should learn through basic physical education.
The QM book contains five sections: first an introduction on general topics (some of which I have just summarized), then technical sections on quadrupedal postures, general movement, crawling,and falling. I might not follow the same order always in my notes, but I'll label each article accordingly.