Once you've determined what sort of photographer you really are (and that may be different from the type of photographer you'd like to be in an ideal world), working through the variables that differentiate digital cameras becomes significantly easier.
Digital cameras come in a wide variety of sizes, and for documentary photographers, small size can be important so the camera fits in a pocket and so the fact that you're taking pictures doesn't overwhelm the event. In contrast, artistic photographers are often willing to carry larger cameras because of their increased quality and flexibility. Most cameras fall in a middle ground, so it's best to hold the camera to see how it feels in your hand before buying it.
Within the form factor that matches your shooting style, you should usually try to get the most megapixels (larger cameras have larger CCDs and can thus capture more pixels). But lens quality makes a difference too, and the best way to determine lens quality is to read detailed camera reviews. Artistic photographers need to pay the most attention to quality issues.
For documentary photographers, this mostly comes down to whether or not the camera has a decent optical (ignore digital) zoom for capturing far away events. For artistic photographers, zoom capabilities are important for those times when you simply can't (or shouldn't) get close enough to the subject (such as a grizzly bear). Also important are macro capabilities for stunning close-ups of flowers, insects, and other small objects.
Extra features may be important: documentary photographers may appreciate a short movie capability, whereas artistic photographers might look for the capability to use an external flash or different lenses. Also pay attention to factors like the camera's battery type. AA batteries are cheap and easily found, and you can buy good rechargeable batteries, but the smallest cameras generally have their own battery packs and chargers.
Everyone has a budget, and here's where you must decide how much to pay. Documentary photographers will pay more for truly tiny cameras with decent quality, whereas artistic photographers pay more for quality, manual controls, and support for additional lenses. Although more money will buy a better camera, you can take good pictures with almost any camera.
Some cameras are easier to use than others, which is important for documentary photographers who need to be able to set options quickly before missing a particular shot. Artistic photographers care more about control, so the camera interface should help them twiddle manual settings easily.
Digital cameras can be quite slow, with a few seconds to start up, a lag between when you press the shutter release and when the picture is taken, and a few-second lag between shots. Documentary photographers in particular should look for faster cameras, so as to avoid missing that perfect shot due to camera lag. Artistic photographers interested in action shots should also pay attention to speed or risk losing great photos.