Modern digital cameras offer the convenience of immediate feedback, more editing power than you’d get in an old-fashioned darkroom, and there’s no film to develop.
Decide if you want a point and shoot camera or a digital SLR. A point and shoot camera is compact (many will fit in your pocket) but the lens can't be changed or replaced. A digital SLR is bulkier, but you can switch lenses "on the fly" to get both very wide angle and extreme telephoto shots. An SLR is more flexible in other ways as well, but if you are mainly interested in taking casual snapshots, you'll probably be happier with a pocket-sized point and shoot camera.
Consider the exposure modes. Almost all cameras come with a fully automatic mode, but as you grow as a photographer or want to get more creative, it's important to also have a Shutter Priority mode (which lets you choose the shutter speed while the camera does the rest) and an Aperture Priority mode (in which you set the aperture and let the camera choose the matching settings). A Manual control will also let you make long night exposure or take shots in very tricky lighting conditions.
Compare the flash modes of similar cameras. The flash should have a red eye reduction mode, and you should be able to turn the flash off or force it to fire so it can serve as a "fill" light outdoors. A "hot shoe" connection lets you attach a more powerful external flash as well.
Compare additional features you might want. Some cameras (especially some premium point and shoot cameras) have unusual modes. A Panorama mode, for example, stitches together multiple photos in one wide or tall scene. An HDR mode combines photos of the same scene taken with different exposures into a single "high dynamic range" photo that has great highlights and shadows -- something otherwise difficult or even impossible to achieve with a normal camera. You might be interested in a camera which includes a fast charger or uses inexpensive replaceable batteries.