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.
If you decide on a point and shoot camera, a key consideration is the zoom range. You can get a "megazoom" camera that has a massive 30X (or more) zoom range, but these cameras tend to be bulky and somewhat expensive. If possible, test the camera in a store to make sure the zoom range is good for the kinds of photos you want to take (wide angle landscapes or telephoto action photos, for example).
Consider the camera's resolution (measured in megapixels). You only need about 6-8 megapixels to make a quality 8-by-10 inch print, but more megapixels will let you crop the photo and still make a high quality print. Cameras come in resolutions as high as 20 megapixels, but you don't need that much resolution to get high quality prints. Bottom line: don't purchase a camera based on the "megapixels" alone.
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 Scene modes. Many cameras, especially point and shoot cameras, have a library of modes such as Landscape, Sunset, Twilight, Snow, and more. These modes are optimized for specific shooting situations. If you like to take casual snapshots, look for a camera with a wide variety of these modes.
Video has become very important even among digital cameras. Check to see if the camera can record high definition video, and if it can record sound. Some cameras also let you do simple video editing on the camera, so there's no need to do it on your computer.
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.
The viewfinder is also important. Most digital cameras rely on an electronic viewfinder or an LCD display on the camera back. Even better: Digital SLRs also let you see directly through the lens with an optical viewfinder. And some SLRs also include a special "live view" mode in which you can see the action as it happens in the LCD display.
If you can test a camera in the store, be sure to evaluate what is known as "shutter lag." You can also read about this in camera reviews online if you don't have the opportunity to try it out firsthand. Shutter lag refers to how long it takes to actually take the photo after you press the shutter release. This should be almost instantaneous -- certainly no more than about a half-second. Older and less expensive cameras have long (and annoying) shutter lag.
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.