If you're shooting video on your phone, you have it pretty easy when it comes to the concept of zooming. Right now, many of them don't offer any zooming while you're recording. I don't think it will be too long before most of them do, and if you want to shoot better video, then you're going to need to think about when you need to use it.
The concept seems pretty simple: if you can't get close, simply zoom in to get closer. But there are many things to consider about what affect that has on your shot.
First of all, the more zoomed in you are, the more shaky your video will be. Especially if you don't have any access to a tripod. The pros have a simple principal: "zoom with your feet". When shooting without a tripod or "off the shoulder" or in your hand in a lot of cases, zooming makes even small amounts of motion from your body (breathing and walking) seem really big. Get as close as you can to your subject and the action to get the best and most steady video. Getting close will even help you clean up your composition.
Now if you're editing a piece and need to get that "wide shot", walk further away. Zoom with those feet. Don't be lazy.
From a practical, real world stand point, we have to consider some extreme situations. Let's say you want to get some footage of the concert you're attending, but you don't want it to look like and earthquake. You also are pretty far away, so you're going to need to zoom to get a good feel for what's going on. Do your best to find a way to steady the camera as much as possible to get those shots.
Secondly, we've got to go back to the principal of shooting as the eye sees. This mainly deals with cameras that have the ability to record and zoom and the same time. Now, as a general rule, zooming should be considered an artistic element. It's been used fantastically throughout the history of video, but in most circumstances, using zoom in the middle of the shot can be pretty jarring. And that's what we're addressing here: using the zoom in the middle of a shot.
Time for an experiment: look at your hand...now look at something far away like a picture or painting. Did your vision zoom in? Unless you're Superman, probably not. You just mentally focused on that object. It didn't get bigger. Now (this is for those who plan to edit) you could walk across the room to get a good look at the picture, but that's the only way it's going to get bigger. And this is what I'm talking about when I say shoot as the eye sees. From any given vantage, the average person is going to see one static image. This is why zooming is used primarily as a special effect. Bottom line, try to avoid zooming as much as possible unless you have a good reason.
So you're standing on the street when a high speed chase comes by. You happen to have your phone out and this would be a great thing to share with your friends and maybe the world. You have the option to zoom and as the cars move away from you what do you do? Well it would be impractical for you to zoom with your feet here, so let this be an example of a time when the rules go out the window because the video itself is what's compelling no matter how poorly it's shot. I shot several high speed chases while working as a news videographer and didn't hesitate to use that zoom. Because people are interested in seeing the cars. Always be asking why and you'll probably make the right decisions.
It's important to not confuse zooming with panning. And I'll be addressing the pan next.