Buy the highest resolution camera you can afford but know that a two-megapixel camera is plenty for the average home user. Another thing to look at is how images are transferred from your camera to your computer. I favor any devices, camera or otherwise, that use a USB cable. Many new computers now have USB ports on the front of the computer and you'll even find USB ports on some monitors, making picture uploading to the PC as easy as can be. Look for a camera that comes with a rechargeable battery pack.
Digital cameras tend to use quite a bit of juice.
That means if you rely on standard alkaline batteries, you'll probably end up spending more on batteries than you spent on your camera. So, to make sure your battery doesn't die in the middle of your niece's college graduation, first make sure your batteries are fully charged. Then, keep an extra set of standard alkaline batteries in your camera case for backup. That way, you won't miss any important shots. Finally, consider what storage medium the camera uses.