The ISO rating of a film is determined by the exposure required to produce a negative (or positive, in the case of slides) with a particular contrast level when the film is developed according to a standard recipe and process (time, temperature, etc.).

Simply put, "push processing" is developing a film for longer than normal; "pull processing" is developing it for less than the normal amount of time.

There are a number of different reasons why you might want to push or pull film. One of those reasons is that sinking feeling you get when you realise after removing the film from your camera that you've shot the whole roll (or at least the last few shots) at the wrong ISO. Pushing or pulling may be able to salvage usable shots. But that's not the ordinary reason for push/pull.

Push processing is often used with high-speed film to shoot at very low light levels. It's not so much that pushing is the best way to go about things, but it's often the only option since the film speed you need simply doesn't exist. You are forced, therefore, to underexpose and overdevelop the film you can get in order to capture the image you want.

Push processing tends to increase grain (or the size of the dye clouds in a colour picture), so it is often used (again with high speed film, which tends to be grainy to begin with) to produce a "grainstorm" effect in an image, giving the picture a sort of pointillistic and "artsy" feel.

When photographers specify a push or pull of less than a full stop, it's usually because they've tested the film and found that it gives the contrast (and, perhaps, the saturation or detail) they want with something other than the ISO standard method. (Pushing and pulling are core components of the Zone System; the idea being that you know the film and its response to developers well enough to get the contrast and tonality right on the negative so that you don't have to monkey around with paper contrast.) I might find, for instance, that shooting ISO 100 chromes at an exposure equivalent of 64 (a 2/3-stop overexposure) then processing for a 2/3-stop pull might be the best way to tame the high contrast in my landscapes on sunny days, and doing the opposite might add some sparkle and contrast on overcast days with flat light.