Almost every digital camera today comes with a zoom lens, one that lets you adjust the length to include more (wide angle) or less (telephoto) of the scene. Most of these cameras have lenses that provide a 3x or 4x increase in lens length. If your camera lens extends from 24mm to 100mm, it's providing a 4x zoom (100mm/24mm = ~ 4x). Other models sport 10x or even 12x zoom lenses.
In both these cases, the difference in how much the lens zooms is determined by how the optical pieces of glass in the lens are adjusted. So this is called "optical zoom" because it's based on the optics of the lens itself.
Many compact digital cameras also have a feature called "digital zoom." (Digital SLR cameras don't have this feature.) This produces a "super telephoto" effect, you might say. Using digital zoom makes your photo look as though the camera has an even longer telephoto zoom. But this is an illusion!
Digital zoom is actually not an effect of the lens at all. It is software in the camera which crops out the center part of the picture and enlarges it in the camera to make it appear as though you used a longer zoom. This sounds neat, you say! How do I use it?
Well, let me caution you. As good as this sounds in theory, you might not like the actual results. Below are two photos of the same scene. The first I made using the maximum optical zoom on the camera. If you enlarge this on your screen (just click on it for a bigger version), you will see that everything looks nice and crisp.
Optical zoom photo
In the second photo, I backed up much farther and used the digital zoom to frame the same thing. If you enlarge this picture on your screen, you'll notice that details are blotchy and edges are smudged. This is the result of the in-camera crop and enlarge process.
Digital zoom photo from farther away
You can almost always get a much better cropped version of your photo by doing it yourself, with photo editing software, than you can by using digital zoom. So I recommend that you test out your digital zoom and see if the quality is acceptable. If you decide it's not, then refer to your camera instruction manual for how to turn off this feature. Your photos will be sharper as a result!