How many pixels
For smaller camera you need at least 4 Megapixels. It is enough to print 11x8.5'' photos with a good quality. If you buy an SLR digital camera I would recommend you 8 or larger Megapixel cameras: larger prints and more options for picture cropping.

Ignore Digital Zoom
Do not use digital zoom. Your computer makes this job better. Only optical zoom provides for a true zoom-in or zoom-out on your subject.

Media Type
It is not very important but I would not buy cameras with CompactFlash media cards. It is very easy to break down small contact pins when you insert or remove the card.

Shutter speed
If you are going to take night scene pictures or make smooth running water effects your camera must allow for long exposures of 1 or 2 seconds. To take pictures of sport events you will need shutter speed of 1/500 or 1/1000.

Zoom
If you buy SLR you do not care much about zoom range. You can buy lenses to cover many zoom ranges. For smaller digital cameras 4x zoom is more than enough. A larger zoom range usually gives bad picture quality, particularly at the extreme zoom positions.

Lenses
It is very important for smaller digital cameras. A poor lens is not sharp, exhibits chromatic aberration (you will see colored fringes at high contrast edges) and barrel effect (straight lines like the horizon line or door frames are curved).

Blur effect for bad lenses

Barrel effect

You can easily check lenses of your camera in the store: zoom out look at the LCD screen and find some vertical line. Pan the camera to get the line closer to the edge of the screen. If you see a curved line - do not buy this camera. You will hate it in a month.

Aperture
The aperture of a lens is the diameter of the lens opening The bigger the aperture, the more light is gathered, and the less light you need to take a good photo. Lens aperture is measured in f/numbers, such as f/2.8, f/3.5. Smaller numbers mean bigger lens openings. I would look for cameras with f/2.8 or smaller.

Special functions
You can read that a camera has many special digital effects: color balance, contrast, increased sharpness, etc. Do not care much about it. Your computer will make this job. I never use these options.

Spare battery
Do not forget to buy a spare battery. Unfortunately, this is a must.

Conclusion:
Do not buy a very cheap camera. Usually it has a very bad lens. Do not look for a large zoom range - you will get a bad lens. Check your camera in the store for the barrel effect. Start with a compact camera: you will use it even after purchasing a more expensive one.