Things to look for when buying a digital camera are:

Zoom - Optical zoom is how far the lens actually zooms in, whereas digital is where the image is cropped and stretched. If a camera says 57x zoom, it's more likely 5.7x optical zoom and 10x digital zoom.

Megapixels - Don't be fooled by the number of megapixels a camera has, unless you want to print big pictures (larger than 12x8") then you won't need anything above 8mp. My £300 SLR (Nikon D40) is 6mp.

ISO - Cameras sometimes advertise high ISO numbers. This is how sensitive the sensor is, and how much detail is captured. At night when there isn't much light, the camera sets a high ISO, although the pictures are very grainy. A camera with ISO 100-1600 is very good.

Lens - I like cameras which have a wide angle lens (28mm on an old film camera). You get more in the picture without having to walk backwards.

Brand - Lumix (Panasonic) are slightly more expensive, but you get what you pay for. With Canon you pay more for their advertising, so I wouldn't recommend them.

Face Recognition - New software in cameras has the ability to pick up peoples faces, and focus accordingly, so you rarely get out of focus shots.

Memory card - An ideal camera would take SDHC (SD Cards 4gb and above).

Personally I don't like small compact cameras because they're easy to drop, quite often don't have a viewfinder, and don't give as good pictures as a DSLR (the big black ones with sticky-outy lenses).