Digital zoom is a method of zooming on a digital camera either by increasing the size of the pixels in the image or by interpolating between them. The image doesn't physically get any closer and no extra information is collected, as the optics on the camera stay in place.
The process on some cameras is equivalent to cropping the image: Only data from the centre of the available image is stored, meaning the resolution or megapixel rating of the photograph is reduced, with a corresponding decrease in file size. The resulting image is identical to a photograph taken with no zoom, except that the unwanted edges of the image are discarded.
Digital zoom is closely related and works similar to how zooming works in software such as Adobe Photoshop. Digital zoom can reduce the quality of images significantly, and as such, is not recommended if the picture is destined to be printed out. Most digital cameras on the market today carry this feature and some rely entirely on this zooming technique and do not have any optical zoom capabilities.
Digitalzoom can reduce the quality of the resulting image, and as digital SLR cameras generally strive for optimum quality over ease of use, such a feature is rarely included in these cameras.
Digitalzoom is just another type of post-processing, and any in-camera processing will be of lesser quality and flexibility than what can be achieved in software on your home computer.
Digitalzoom is simply a software program running in your camera that enlarges your original image (as if you were zooming in with a real optical zoom lens).