The ISO denoted on a box of film is not gospel. In fact, some photographers just consider that number, whether it be 100, 400, or 800, a starting point. Thanks to push processing, you can shoot ISO 800 film, for example, at 1600.
As explained quite well in the article, How to Push and Pull Film, "...which is essentially shooting your film at the wrong speed and then developing it at that same wrong speed. To 'push' your film is to shoot and develop at a faster film speed, while to 'pull' film is to shoot and develop at a slower film speed. This process will change elements of your photograph such as grain size and contrast, but can also be a life saver in more than one tricky situation."
You may have seen this option on the order form for your lab. Above is an example from a lab I use, Photoworks San Francisco. As you can see, there is a cost associated with push processing, because the lab has to handle the film separately.
Generally speaking, the ceiling for push processing is 3 stops. So, for ISO 400 film, that means you could shoot it at ISO 3200, then instruct the lab to push-process 3 stops. Obviously, the images will look different than when shot at the film's native ISO. You'll see a more pronounced grain pattern, colors my shift, and middle tones may be compromised. Then again, if you want that drama, pushing is a good option.
The instance where I use it most is to correct a mistake on my part. If I have my camera's meter set to ISO 400 when I've loaded ISO 200 film, pushing is a lifesaver. Instead of getting back a roll of underexposed images, I can recover from my error by telling the lab to push that roll one stop.
If you process at home, you have even more control because you can tailor your development to you shooting style. When I shot high school sports with Tri-X, I had my own push recipe with Acufine developer that allowed me to shoot at ISO 1000.
The most important thing about push processing is knowing that you can do it. When you're in a situation where more ISO is the overriding factor, then go ahead, jack it up..