Cartier patented the first "folding buckle" in 1909. Called the "boucle déployant" in French, this would be exclusive to Cartier for decades, and was used on the first wristwatch for me, the Cartier Santos starting in 1911.
In contrast to a simple buckle or folding clasp, a deployant has two or more hinged metal components that fold over each other to close the watch strap or bracelet. Many deployants are designed to resemble conventional buckles, but some have subtler design, hiding the clasp entirely. Many deployants have one or more button to release the catch but others are purely friction attached.
Most deployants reverse the classic strap attachment, with the longer "tongue" portion attaching at the top of the watch case and the shorter "buckle" portion at the bottom.
A single deployant has a single hinged section that folds over underneath the strap or bracelet and a clasp that attaches at the end. Single deployants are simple in design but are sometimes cirticized for their bulk and for the fact that the visible clasp sits at one end rather than neatly under the wrist.
Many single deployant clasps can be attached to a conventional pin buckle watch strap. Some watches offer an option of either a pin buckle or deployant, depending on price or model, with the same basic strap. Sometimes the strap is reversed to accommodate the deployant, but others work in the regular strap orientation. Some manufacturers have developed deployants that fold the longer "tongue" of the strap on the inside but most have conventional loops to hold the tongue.
A double or butterfly deployant has two hinged sections, one at either end of the base component. These attach at the middle of the wrist and are sometimes more compact.
This type of clasp usually cannot be used with a conventional strap and requires a specially-made piece with spring bar attachments on both elements.
Similar to a single deployant, a tri-fold clasp has a single folding arm inside the strap or bracelet but adds a large overhanging plate outside as well. This outer portion has a spring bar or simple bar to attach to the folding component. Some tri-fold clasps also feature a second "safety" catch which folds over the outer plate to keep it from unlocking.