In photography, reciprocity is the inverse relationship between the intensity and duration of light which determines the reaction of a material sensitive to light.
Reciprocal failure is how we describe the film reacting unevenly to exposure. Usually, film exposure is quite linear: exposing film at f/2.8 for 1/60 s will give you the same negative density as exposure at f/4 for 1/30 s or f/2 for 1/125 s. However, when you start to reduce the number of photons hitting your photosensitive material per second, things get a little out of sync. You need more photons to make the photosensitive material react, so you need to extend your exposure time.
In practice, this means that for long exposures you need to calculate a reciprocity failure. The length of time depends on the type of film you are using. With black and white, it's quite simple: you just have to extend the times. With color film, because different layers lose their reciprocity at different rates, you will often get a color change and you will have to correct this with color filters. This is one aspect of photography that digital does better.