Digital Zoom versus Optical Zoom
As compact digital cameras started to become popular several years back, one of the ways manufacturers attempted to differentiate their products was by extending the zoom range. After all, the lens wasn't interchangeable as they are on SLR cameras, so if they could offer you more zoom for your buck than the next guy, you might just buy from them instead. But there are limits to what lenses can actually do, so someone came up with the idea of digital zoom. Don't fall for it, unless you really understand what you are getting.
First, if you take the longest focal length a lens is capable of and divide it by the shortest, you get the zoom range. Expressed as a magnification power or "x" factor, compact camera lenses generally fall somewhere in the range of 2x to 5x. There are guaranteed to be compromises if they stretch things too far though, so 5x is about the limit of what is out there. Even then and the lens will probably suffer from at least some degree of barrel distortion at the wide and pincushion at the extreme telephoto end. Maximum lens aperture will generally decrease as you zoom a lens out longer too, but the limiting factor tends to be distortion rather than image brightness.
By changing lenses on an SLR camera, you can extend zoom range well beyond what is possible with a point and shoot. Without resorting to fisheye lenses or teleconverters, I can go from all the way from 12mm focal length out to 400mm, an amazing 33x zoom range. And lenses with even longer focal lengths are available, albeit for a price.
So to simulate a longer zoom range or merely to save costs in the optical design, most small point and shoot cameras supplement what they can really do with what they call digital zoom. What this really is though is just a simple cropping mechanism such that the image formed is created by only the central portion of the CCD or CMOS sensor. No matter how many megapixels your camera may have, it has less when you resort to digital zoom. You can do the same thing yourself on your computer once you get home. You can even keep the full image and still isolates any part that interests you as a second copy from the same image.
But while image quality suffers, the appearance of digital zoom in terms of perspective is identical to that of real zoom since perspective depends solely on relative subject distance, not on focal length. It's a common misconception that wide angle lenses expand and distort perspective while telephoto lenses compress it, but they don't really. It's just that you generally get up close to your subject when using a wide angle lens and you generally reach for a long telephoto when you are far away. Both optical zoom and digital zoom narrow the angle of view without altering perspective. The difference is that the entire sensor gets used with optical zoom, but only an increasingly small central part of it does with digital zoom.
In most cases, digital zoom still results in an image of the same size since the camera resamples the image to scale it up to the number of megapixels that a regular image has. Whether this is done to fool the user or for some other undisclosed reason, it means your digitally cropped images will fill your card just as fast as normal and take just as long to get their as uncropped ones do. A few compact cameras come with an option to similar to Nikon's DX crop mode used in the D2x and D3 that lets them actually create a smaller image from the central part of the sensor. With this approach, a five megapixel camera might put out a three megapixel cropped image to save time and space. If you are in need of either, this may be a viable option.
It's also worth considering that any resampling done during digital zooming happens before the image is saved as jpeg data. If you heavily compress your jpegs in-camera and then enlarge them on your computer once you get home, the resulting artifacts will be more evident than if you employed digital zoom to get the same image cropping. But if you care about the quality of your images, a better option is simply not to compress them so much. As a general rule, cropping your images after the fact on your computer is a better option because of the flexibility and control you gain.
Regardless of whether you do decide to use digital zoom though, at least realize that it is not the same as optical zoom. It's a compromise that definitely doesn't come for free.