The Zone System is a way of thinking about the range of tonal values in a black and white print and all the decisions a film photographer has to make from the point of conceiving of the image to the print itself.
The system begins by defining 11 tonal zones on the print; from 0 representing maximum black of a print, 10 pure paper white, and 5 resting at the center between the two. Each zone represents a halving or doubling of light. Subsequently we can consider the qualities of a film negative needed to create a print. Notice that you could similarly split the negative into 11 zones, similarly representing halving or doubling of light. Lastly when we consider creating a negative, we most often talk about operating our camera in stops. Each aperture value (2.8, 3.5, 5.6, ...), each shutter speed value (1/30, 1/60, 1/125, ...), and each ISO (100, 200, 400, ...) representing again a halving and doubling of light. This cohesive relationship between the light to expose the negative, and the light to expose a print, provides a means for photographers to visualize the resulting print at the point of conceiving of the image.
For example we can measure the light at a point in the scene and then use the zone system to visualize how our print will change depending on how we were to expose the negative. When we measure light we use a light meter. Light meters are designed to tell us the exposure settings needed to create a zone 5 (middle grey) tone on the negative (that will create a zone 5 grey on the print). If we take a spot meter and measure a part of our scene and consider that as middle grey, we can then measure other parts of our scene and visualize what they might look like on the print. We might find that the sky is 6 stops brighter than our initial reading, such a value when printed would be beyond zone 10, and we know that it will render only as pure white; or we might find that the darkest shadow in the scene is 6 stops darker than our initial reading and we'll know that it will only ever be pure black.
At the heart of zone exposure is the idea of our role as photographer in choosing the exposure and the impact that will have. As above we can measure a spot in our scene, but we are not beholden to our meter. We could expose to represent that as middle grey, or just as easily we could stop down our lens and place that value at zone 3 or conversely open up and place it in zone 7.
This picking gives us some helpful tools for day to day usage. It is common to "protect the shadows" by measuring the darkest shadow that the photographer wants textural information in, and choosing to place that in zone 3 (stopping down 2 stops). But it's important to recognize that as photographers we can pick to place that value anywhere and use the system to understand how the print will look.
At the heart of zone exposure is the ability to adjust the scale of tonal values of the negative (contrast) by increasing or decreasing development. As a negative is developed longer the highlight values continue to increase in density up until it reaches the maximum density of the film. Simultaneously the shadow values will increase in density much more slowly up until it exhausts exposed silver in the negative. Using this observation we can modulate our development to expand our contract the zones represented in the film.
If for example we had a scene where we measured only 3 stops between the darkest and the lightest parts of our scene and if we were to expose and print that negative with a grade 2 filter we might be disappointed with the relatively flat image produced with little range between the shadows and the highlights.
Conversely if we had a scene that measured with 7 stops between the darkest and lightest parts of our scene and we were to expose and print that negative with a grade 2 filter we might be disappointed with the print, noticing that detail that we saw in the shadows or the highlights was missing in the print, falling instead into pure black or paper white.
We can control the way the print renders by adjusting the negative through expanding or contracting development time. Lowering development time will reduce contrast and increasing development time will increase contrast.
In order to know how exposure and development will impact the print we need to test the changes. While we can get suggested starting points from places like The Massive Dev Chart or from the film or developers data sheets, the reality is that there are many variables that exist only in our domain as photographers. There are variances in the accuracy of light meters, accuracy of shutters, accuracy of thermometers, differences in agitation, and on and on.