I've delayed responding here to try to make sure that I at least think that I understand how the zone system works (even though I've been using it at some level for my LF work for a while).
Originally it was devised for B&W negatives, and although Adams did shoot colo(u)r slides, which are much more similar to digital, I've never seen any description of how he managed exposure there.
Essentially the "Zones" refer to the latent image on the undeveloped negative, the developed negative and final print have "Values".
Each point in the scene has a luminance, which is mapped by choice of film, exposure and aperture to a zone, thus a given part of the image may be placed on a particular zone and all other points fall onto zones in a linear fashion. The mapping to values is then controlled by development and printing. To aim with the zone system is to generate developed negative that are relatively easy to print.
It is probably easiest to understand if we consider setting the exposure for the shadows (which is normal for negatives -- slides and digital are better metered for the highlights).
Suppose the shadow region reads 8 on a spot meter, and we want this on Zone II (detail just visible), then to get the exposure we would read off the exposure for 11 on the meter (the meter reads for Zone V, 3 stops brighter than Zone II).
Now consider the highlights which we want to have Value of VIII. If they read 13 on the meter then they will only be on Zone VII, which with normal development will give a Value of VII, therefore to move the value to VIII we need longer development (in Adams' terminology N+1) this will have little effect on the shadows, in more typical modern parlance this is a 1-stop push. Similarly if they meter at 16, then the development must be reduced to move Zone X to Value VIII (N-2 development or a 2-stop pull).
I think the biggest difference with digital is the existence of a hard saturation limit, beyond which NOTHING can be recovered, although even with film is is possible to get completely blown highlights (see Ansel Adams' "Martha Porter, Pioneer Woman").