When people push film they are taking a film like HP5 at 400 ISO and shooting it as if it was ISO 1600 (pushing the ISO). If they processed it normally the pictures would be two stops underexposed. So they leave it in the developer longer so the film will print better. It makes the images a bit contrasty but there is no choice because the subject is so dark. Most people push ISO 400 films to about 1600 or 3200. Quality is greatly reduced while contrast is increased. They do this because of a low light situation.
Some photographers use this technique on a low contrast day (raining) to get the images a bit more contrasty while printing. They might push the 400 to 640 or 800 and live with a slight quality loss. I've done a fair amount of this and Ansel Adams made a science out of it. His pictures are generally considered better than mine as well. If you don't know St. Ansel's work, you better quit photography or check this.
Pulling a film is not as common, although I use it all the time.
If you take a 400 ISO film and shoot it at ISO100 the negatives will be too dark to print properly. So you cut back the development time by about 30% which reduces the contrast. On a super contrasty day this technique allows you to control the contrast while still getting great detail overall and more detail in the shadows. I like full shadow detail which this technique provides.
I usually shoot my ISO 400 film at 200 and reduce the development slightly to get a negative that prints nicely on my enlarger and still retains great shadow detail. It would look darker than most people but that's how I like it.
People shoot large format negatives because each frame is it's own piece of film and can be developed individually for the highest quality. Wow, develop each frame separately, these people should discover digital.
If you have never pushed film what are you waiting for. Buy a used film camera that uses the same lenses as your digi. If you don't know what to get, then get a Leica M6 and a 35mm lens. Now buy some film and give it a whirl, you might learn more about photography than you thought. It's harder than it looks but a ton of fun. And you might get a great shot that rivals Ansel's halfdome, not likely but it could happen.
My retail store for traditional photo supplies is www.bigcameraworkshops.com
My retail store for rangefinder gear and Zeiss lenses is www.rangefinderstore.com