Beyond the top three factors, most consumers usually diverge in terms of what they look for. The following are things to consider, but not necessarily in the order I choose to lay them out.
Optical zoom - This ranges from 1 x (no zoom) to 10+ x and describes the amount of visible field that you can isolate with the lens. No zoom means that you're stuck with what the viewfinder sees unless you move closer or further away. Doubling the zoom (2x) means that an object will be twice as tall and wide on 2x as the 1x version. A common mistake is to confuse specifications for optical zoom with digital zoom, a method of zooming which simply involves cropping into the image, enlarging to the normal size, and the loss of resolution. Sometimes camera manufacturers will advertise, for example, a 12x overall zoom, with a 4x optical and 3x digital zoom. Since cropping and enlarging is something you can always do on a computer later on, the optical zoom will be the number to look at.
Size - The camera's size is something that will affect where you can bring it and how convenient it will be to bring along. If you expect to be travelling, you may want to consider that larger cameras make easier theft targets. Smaller cameras, while being very portable, also tend to have less features because there's less room for them. As technology is improving, smaller cameras are evening out in most fields of comparison, except optical zoom, which depends on physical parameters.
Shape & Layout - This is something that a lot of people consider without even thinking about it. How a camera feels in your hand is vital to how much you enjoy shooting with it. It shouldn't be too big, too small, too heavy, or even too light. Even if you don't intend to buy your camera in a store, you should compare the feel you get from your top choices before purchasing. Examine especially how all the buttons, dials, and switches are arranged on the camera body. Do you turn the camera off when you were intending to zoom? Do you press the shutter button instead of the power button? The layout you are comfortable with usually depends on what you're used to, from older cameras you've used.
Scene Modes - Scene modes are preprogrammed settings that are tailored for a specific type of shot. Of these, the most common are macro, portrait, landscape, sports, and night portrait. Scene modes are an easy way to take a shot without having to know how to change the settings to get the best results.
Macro is a setting that usually only works at the lowest zoom (or widest angle of view). It allows you to be able to focus on a subject that is just inches away. It's so great for flower shots that that is usually the symbol that this mode is given.
Portrait modes usually set an auto flash with red-eye reduction. It determines the exposure near the middle of the frame (where the face should be) and uses a larger aperture to focus more attention on the subject.
Landscape modes use a small aperture to get as much of the picture in focus as possible. The flash is set at off because the flash won't be able to reach most of the landscape anyway.
Sports mode lets you take shots at the fastest shutter speed allowable by the amount of light.
Night portrait is the same as portrait except that the flash will go off at the end of the exposure instead of the beginning. This allows more light from the background to hit the digital sensor before capturing the subject.
Other Features - Cameras can come with many features that you may not need on the camera itself. Some offer a panoramic mode that helps you piece together adjacent shots. This is very handy if you love landscapes. You may be able to find black and white and sepia modes, which aren't unique if you can also do that on your computer. Cameras with a focus-assist light are helpful if you take many pictures in low-light conditions. The key to a lot of these extra features is figuring out which ones you'll use and which ones you're just paying money for.
Attachments - Higher end point-and-shootor SLR-like digital cameras feature a threaded lens that allows you to attach wide angle adaptors, tele-photo converters, filters, and other accessories to the lens. These parts are usually sold separately and can be quite costly, but, if you want to be more creative with your pictures on the camera, they are a good investment.
There are many other things that you can consider, and I've only listed the most common ones. I could write forever about how I would pick my next non-SLR digital camera, but that would give you too much to read. Besides, my next digital camera will, I hope, be an SLR!