We never really got into it beyond the name: You fold space to "jump" from one place to another. There was a "fold space" we used a couple times, kind of a place between instances where time's lapse was a question of perspective; A ship makes an "instantaneous" jump from point A to point B, but a person's consciousness and physicality could get hung up there indefinitely under the right circumstances.
The implication late in the series was that folds could only happen within the Briar Patch or connected environs. That gets tied up in the plot of the series, so there are some spoilers in going into why fold jumps and the Patch are interconnected.
A "fold drive" wasn't a driver in the sense that a "warp drive" is: It's not a powered reactor feeding equipment that generates a field. It was an harmonic amplifier that used energy fed to it on a subspace level to open an event horizon and allow for that "instantaneous" jump between coordinates.
A warp drive "warps" space around a vessel, changing the laws of physics, essentially the speed light travels, but submersing the vessel into subspace. Higher warp factors, lower levels of subspace, higher speed of light (and smaller universe).
A fold drive doesn't "warp" space. It opens a kind of singularity into a "fold space" (an extremely deep level of subspace) that allows "instantaneous" transit.
That's the way I thought about it when I conceived the episodes I wrote. Other writers may have used different mental gymnastics.
Dan