Ray depth is the minimum number of reflections or refractions a ray is allowed before it is terminated. After the ray depth limit, a random number of extra reflections or refractions may occur, each with a 50% chance of termination. This technique is called "Russian Roulette" and is used to reduce the variance in the path traced image.
The effects of ray depth are most noticeable when rendering scenes with many adjacent transparent or shiny materials. Compare the images below. There are two layers of glass blocks in the lighthouse. If the ray depth is too low, the average light ray can not pass through all layers of glass block before being terminated. Besides ray depth all other parameters were unchanged in the below renders.
This image was rendered using the default ray depth (= 5). Russian roulette allows some rays to pass through all glass blocks, but most do not make it through:
This image was rendered using ray depth = 10: