Stops are all about how much light you capture with the camera. One stop more is twice as much light. One stop less is half as much light.

What does "stop" mean in this context, and, looking at an image, how would I determine how many stops there are between a bright section and a dark section?

When something such as "... the background is two stops darker than the key light..." is said it is usually in the context of metering the scene, not looking at a completed image. If it is said in reference to a finished image it is being used more as shorthand to say that there appears to have been a certain amount less light on the background than on the area illuminated by the main light. Although it is possible to use software tools with digital images or even colorimeters with a physical print to measure the exact difference, in normal practice the measurement is done of the scene itself, not the result.

When speaking about stops in terms of light sources or different areas of the scene, one is referring to the difference in exposure value needed to render each area of the scene at an exposure level that would properly expose the same object (e.g. an 18% gray card).

Assuming constant ISO and shutter speed, if your reflective light meter pointed at the area directly under the key light says use f/8 and the same light meter pointed at the background says use f/4 then you can say that the background is 2 stops slower/darker than the key light. This is because f/4 is two stops slower than f/8. If you're using an incident light meter the same would be true if you place it under the key light and get a reading two stops brighter than when you place it under the light falling on the background.

If you are using your camera's built in light meter, which measures reflective light, use the smallest metering circle available. It is usually labeled spot metering and will cover anywhere from 1.5% to 3% of the frame on some of the higher end models and about 6% to 9% of the frame on lower models. Be sure to place the spot over the area you wish to meter in your viewfinder.