Pushing film is nothing more than setting your camera's ISO higher and developing the film for longer periods of time. The benefit being slower speed films (ISO 100, 200, etc.) can be shot at faster shutter speeds (ISO 400, 800 and 1600). This is helpful when you're shooting film when there's less light available, such as sundown or at night. One thing to keep in mind, however, is that pushing film requires you to push the entire roll. The decision to push film must be done before you take the first image a roll of film.
Visually, pushed images render with more contrast and grain. I'm a fan of pushing black and white film; I find it renders images with very raw and real look. Below are a few examples: