Here's what to look for in a digital camera: resolution, lens aperture, and zoom range. The resolution determines the quality of the prints you will get from your camera (the more megapixels, the more clear your larger prints will be). The lens aperture refers to the opening of the lens and determines the amount of light needed to take photos. If you take a lot of indoor photographs, and prefer to do this withou the flash, you will need a higher f/number (the measurement for aperture. The smaller the f/number, the more light taken in through the lens. The lens zoom range is self explanatory: the higher the zoom, the greater the depth (and width) you can achieve with your camera. Average camera zoom length is 3X zoom.
What to look for in a digital camera would of course be determined by the level of photography you intend to use it for. The main things to consider would be the resolution capabilities, zoom range, and lens aperture.