I've been asked to implement an edge-based disparity map, but I fundamentally don't understand what a disparity map is. What is the definition of a "disparity map"?
Disparity map refers to the apparent pixel difference or motion between a pair of stereo images. To experience this, try closing one of your eyes and then rapidly close it while opening the other. Objects that are close to you will appear to jump a significant distance while objects further away will move very little. That motion is the disparity.
In a pair of images derived from stereo cameras, you can measure the apparent motion in pixels for every point and make an intensity image out of the measurements.
See this for an example. You can see the objects in the foreground are brighter, denoting greater motion and lesser distance.