Don't be afraid if you always felt confused by the concept of pushing or pulling film, because it's quite normal, since it's not intuitive as many people stand. Yep, it's a matter of overexposing or underexposing a certain film, but still it's not so intuitive understanding how to take advantage of these ways of shooting film. It's not easy to manage how to learn it, seeing all these people commenting "TriX 400 shot and developed at 1600" or "Kodak Portra 400 shot @ 100"; this kind of approach doesn't help so much.
You’ll need to pull your roll when you only have a 3200 ISO film and you’re shooting an outdoor session at bright daylight. In this situation the camera only goes to very high speed and no-one is happy to have not the freedom to choose the aperture you think it’s better (sometimes is awful to shoot at F/16). So you can shoot the roll like it’s 400 ISO and the ask the lab to pull.
The lab is going to develop that particular roll for less time, corresponding to the time needed for 3 stops less than the labeled 3200 ISO. Pulling is less common since it’s really not that frequent to have a higher ISO film than you wished; moreover, modern film rolls (like all Portra) have a hugely wide dynamic range, so the still look great even when overexposed by 3 stops.
The typical situation in which you should take advantage of pushing is in low light conditions, for example when you are the one in charge for photographing a party at night. In the external locations, a Portra 400 could be the best choice ever, that's right; when the ceremony starts, a Portra 400 roll is not the best because of the very low light conditions inside that environment; once outside, the party could start after the sunset, when a 400 iso of your beloved Portra is still not enough. The risk of blurred pictures lets you understand that you need a different film, but is unpleasant to travel with tens of different roll types in your bag.
In these moments, pushing is the only solution you have to prevent blurred images (because you could be the best photographer ever, but everyone has hands shake). When your lightmeter says to shoot at 1/15, you cannot let the risk win. The solution is quite simple: use your 400 speed film as if it was 1600 iso, setting the lightmeter at 1600. Those 2 extra stops would make such a difference in those situations, letting the lab know that you used your film at 1600, as an off-label indication. Needless to say you have to mark that on the roll somehow, so that it won’t get mixed up with all those Portra 400 you shot before the ceremony at their regular 400 ISO. The lab will consequentially develop longer than the not-pushed Portra 400.
Pushing means chemically extend the ISO of your film roll, previously underexposed. In the previous example, the film roll was underexposed by 2 stops, making it an "imaginary 1600 roll". The lab will then chemically compensate for this off-label use of your film roll. This management of the iso started an endless misunderstanding about underexposing, because usually we all overexpose by 1 stop in order to get a better final result, but pushing film is a completely different story. We all know that color negative film looks great when overexposed, getting all those creamier skin tones with more detail on the shadows. While pushing, you could still overexpose half stop, your behavior should be the same.
However it must be clear that pushing film has its own “side effects”: you’ll assist at an increase in contrast/grain and a general detail loss in the shadows. Some films lend themselves better to pushing, like all Kodak’s Portra. Moreover, pushed color negative film results vary a lot, even with Kodak Portra 400, the colors can get a little funky when pushed at 3200 ISO.