In short, reciprocity failure is how we describe film reacting unevenly to exposure. Usually, film's exposure is pretty linear: exposing film at f/2.8 for 1/60s will give you the same negative density as exposing at f/4 for 1/30s or f/2 for 1/125s. When you start to reduce the number of photons hitting your photosensitive material per second, though, things get a bit out of sync. You need more photons to get the photosensitive material to react, so you need to extend your exposure time.

In practice, this means that for long exposures, you need to calculate in a reciprocity failure. How much for how long depends on the type of film you're using, here's an example chart for Kodak films. With black and white it's pretty simple: you just extend times and be done with it. With color film, because the different layers lose reciprocity at different rates, you'll often get a color shift and you need to correct for that with color filters. This is easily one aspect of photography that digital does better.

One thing worth remembering is that reciprocity failure is a localized effect, not one across the frame. In certain situations, it can introduce a sort of contrast boost: when your shadows aren't reflecting enough light for proper exposure, while your highlights are, the shadows will be darker than they are in reality. Many guides for handling rep failure will tell you to pull the film in development.